%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET016-7 : TPTP v9.3.1. Bugfixed v7.3.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n008.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 08:28:52 AM UTC 2026
% Result : Unsatisfiable 141.90s 31.91s
% Output : Refutation 132.27s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 168
% Syntax : Number of formulae : 856 ( 219 unt; 0 typ; 0 def)
% Number of atoms : 2087 ( 936 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 7193 (1174 ~; 963 |; 0 &;5056 @)
% ( 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 : 1091 ( 0 ^;1091 !; 0 ?;1091 :)
% 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(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/sandbox2/benchmark/theBenchmark.p',operation4) ).
thf(222,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(223,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)],[222]) ).
thf(224,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)],[223]) ).
thf(6,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(180,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(181,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[180]) ).
thf(5810,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)],[224,181]) ).
thf(57,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(291,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(10,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(190,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(191,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[190]) ).
thf(192,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[191]) ).
thf(139,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(483,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(70,axiom,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_set_in_its_singleton) ).
thf(319,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(320,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[319]) ).
thf(321,plain,
! [A: $i] :
( ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(lifteq,[status(thm)],[320]) ).
thf(577,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[483,321]) ).
thf(578,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[577]) ).
thf(579,plain,
( ( omega @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[578:[bind(A,$thf( omega ))]]) ).
thf(969,plain,
! [B: $i,A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,579]) ).
thf(970,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( omega @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( omega @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[969:[bind(A,$thf( omega @ singleton )),bind(B,$thf( B ))]]) ).
thf(1301,plain,
( ~ ( omega @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[970]) ).
thf(23,axiom,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_subclass) ).
thf(216,plain,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(6020,plain,
( ~ $true
| ~ ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1301,216]) ).
thf(6021,plain,
~ ( null_class @ ( omega @ singleton @ subclass ) ),
inference(simp,[status(thm)],[6020]) ).
thf(6060,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[291,6021]) ).
thf(6061,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[6060]) ).
thf(6085,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( omega @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[6061]) ).
thf(140,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(484,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(2,negated_conjecture,
universal_class @ ( w @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_ordered_pair_determines_components1_2) ).
thf(170,plain,
universal_class @ ( w @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(583,plain,
! [A: $i] :
( ( ( universal_class @ ( w @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[170,321]) ).
thf(584,plain,
! [A: $i] :
( ( ( universal_class @ ( w @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[583]) ).
thf(585,plain,
( ( w @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[584:[bind(A,$thf( w ))]]) ).
thf(1027,plain,
! [B: $i,A: $i] :
( ( A
!= ( w @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,585]) ).
thf(1028,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( w @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( w @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1027:[bind(A,$thf( w @ singleton )),bind(B,$thf( B ))]]) ).
thf(1165,plain,
( ~ ( w @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( w @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[1028]) ).
thf(4154,plain,
( ~ $true
| ~ ( null_class @ ( w @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1165,216]) ).
thf(4155,plain,
~ ( null_class @ ( w @ singleton @ subclass ) ),
inference(simp,[status(thm)],[4154]) ).
thf(4185,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( w @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,4155]) ).
thf(4186,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( w @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[4185]) ).
thf(4201,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( w @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[4186]) ).
thf(40,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(254,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(255,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[254]) ).
thf(11,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(193,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(194,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[193]) ).
thf(1688,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)],[255,194]) ).
thf(1689,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1688:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1718,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1689]) ).
thf(88,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(365,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(50,axiom,
omega @ inductive,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(276,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(35,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(243,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(244,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[243]) ).
thf(9467,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[244,6021]) ).
thf(9468,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9467]) ).
thf(9526,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9468]) ).
thf(9564,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[9526]) ).
thf(10603,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[276,9564]) ).
thf(10604,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[10603]) ).
thf(10638,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[10604]) ).
thf(3,negated_conjecture,
w != y,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_ordered_pair_determines_components1_3) ).
thf(171,plain,
w != y,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(173,plain,
w != y,
inference(polarity_switch,[status(thm)],[171]) ).
thf(174,plain,
y != w,
inference(lifteq,[status(thm)],[173]) ).
thf(865,plain,
! [B: $i,A: $i] :
( ( A != y )
| ( B != w )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,174]) ).
thf(866,plain,
! [A: $i] :
( ( A != w )
| ~ ( y @ ( A @ subclass ) )
| ~ ( A @ ( y @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[865:[bind(A,$thf( y ))]]) ).
thf(1259,plain,
( ~ ( y @ ( w @ subclass ) )
| ~ ( w @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[866]) ).
thf(1582,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ~ $true
| ~ ( w @ ( y @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[216,1259]) ).
thf(1583,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ~ ( w @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1582]) ).
thf(1620,plain,
! [A: $i] :
( ( A != y )
| ( null_class != w )
| ~ ( w @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1583]) ).
thf(1633,plain,
( ( null_class != w )
| ~ ( w @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1620]) ).
thf(151,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(509,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(510,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[509]) ).
thf(511,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[510]) ).
thf(138,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(482,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(1608,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ~ ( y @ ( w @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[482,1259]) ).
thf(1609,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ~ ( y @ ( w @ subclass ) ) ),
inference(simp,[status(thm)],[1608]) ).
thf(1615,plain,
! [A: $i] :
( ( universal_class != w )
| ( A != y )
| ~ ( y @ ( w @ subclass ) ) ),
inference(simp,[status(thm)],[1609]) ).
thf(1630,plain,
( ( universal_class != w )
| ~ ( y @ ( w @ subclass ) ) ),
inference(simp,[status(thm)],[1615]) ).
thf(1860,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( universal_class != w )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,1630]) ).
thf(1861,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1860]) ).
thf(1885,plain,
! [A: $i] :
( ( A != y )
| ( null_class != w )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1861]) ).
thf(1892,plain,
( ( null_class != w )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1885]) ).
thf(2987,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != w )
| ( universal_class != w )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,1892]) ).
thf(2988,plain,
! [A: $i] :
( ( A != w )
| ( universal_class != w )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2987:[]]) ).
thf(3399,plain,
( ( universal_class != w )
| ( w @ ( w @ regular @ member ) ) ),
inference(simp,[status(thm)],[2988]) ).
thf(137,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_2) ).
thf(479,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)],[137]) ).
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/sandbox2/benchmark/theBenchmark.p',operation2) ).
thf(249,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(109,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class_definition) ).
thf(414,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(415,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[414]) ).
thf(16,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction2) ).
thf(202,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(203,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ restrict ) ) )
= ( C @ ( B @ ( A @ cross_product ) @ intersection ) ) ),
inference(lifteq,[status(thm)],[202]) ).
thf(2463,plain,
! [A: $i] :
( ( element_relation @ ( A @ ( universal_class @ cross_product ) @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(rewrite,[status(thm)],[415,203]) ).
thf(132,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',only_member_in_singleton) ).
thf(468,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(65,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(308,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(107,axiom,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_is_null_class) ).
thf(409,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(410,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[409]) ).
thf(411,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
= null_class ) ),
inference(lifteq,[status(thm)],[410]) ).
thf(1580,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ~ ( y @ ( w @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,1259]) ).
thf(1581,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ~ ( y @ ( w @ subclass ) ) ),
inference(simp,[status(thm)],[1580]) ).
thf(1616,plain,
! [A: $i] :
( ( A != w )
| ( null_class != y )
| ~ ( y @ ( w @ subclass ) ) ),
inference(simp,[status(thm)],[1581]) ).
thf(1631,plain,
( ( null_class != y )
| ~ ( y @ ( w @ subclass ) ) ),
inference(simp,[status(thm)],[1616]) ).
thf(2164,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( null_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[482,1631]) ).
thf(2165,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2164]) ).
thf(2174,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != w )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2165]) ).
thf(2183,plain,
( ( universal_class != y )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2174]) ).
thf(3266,plain,
! [A: $i] :
( ( null_class != null_class )
| ( universal_class != y )
| ( A != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,2183]) ).
thf(3267,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3266:[]]) ).
thf(3409,plain,
( ( universal_class != y )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[3267]) ).
thf(4623,plain,
! [A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class )
| ( universal_class != y ) ),
inference(paramod_ordered,[status(thm)],[3409,321]) ).
thf(4624,plain,
! [A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[4623]) ).
thf(4648,plain,
! [A: $i] :
( ( universal_class != y )
| ( ( y @ regular )
!= A )
| ( ( A @ singleton )
!= null_class )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[4624]) ).
thf(4653,plain,
( ( ( y @ regular @ singleton )
!= null_class )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[4648]) ).
thf(4958,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( y @ regular @ singleton ) )
| ( null_class != null_class )
| ( universal_class != y )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[411,4653]) ).
thf(4959,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( y @ regular @ singleton ) )
| ( universal_class != y )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4958]) ).
thf(4960,plain,
( ( universal_class != y )
| ( universal_class @ ( y @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4959:[bind(A,$thf( y @ regular ))]]) ).
thf(37,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(247,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(47,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(269,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(1874,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( universal_class != w )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[291,1630]) ).
thf(1875,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1874]) ).
thf(1882,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= y )
| ( element_relation != w )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1875]) ).
thf(24,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one1) ).
thf(217,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(121,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_identified_by_element2) ).
thf(440,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(91,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(372,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(646,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)],[372,321]) ).
thf(647,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)],[646]) ).
thf(648,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[647:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ unordered_pair ) ))]]) ).
thf(655,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[648]) ).
thf(97,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_4) ).
thf(385,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(7,axiom,
choice @ function,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(182,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/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(214,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(215,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[214]) ).
thf(4698,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[182,215]) ).
thf(4699,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[4698]) ).
thf(4700,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4699:[bind(A,$thf( choice ))]]) ).
thf(31,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class2) ).
thf(231,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(232,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[231]) ).
thf(6937,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)],[4700,232]) ).
thf(6938,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class ) ),
inference(simp,[status(thm)],[6937]) ).
thf(6939,plain,
choice @ single_valued_class,
inference(pattern_uni,[status(thm)],[6938:[bind(A,$thf( choice ))]]) ).
thf(168,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/sandbox2/benchmark/theBenchmark.p',property_2_of_ordered_pair) ).
thf(550,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)],[168]) ).
thf(134,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/sandbox2/benchmark/theBenchmark.p',corollary_2_to_unordered_pair) ).
thf(472,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)],[134]) ).
thf(41,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(256,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(257,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[256]) ).
thf(2956,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,257]) ).
thf(2957,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2956:[bind(A,$thf( subset_relation ))]]) ).
thf(10653,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,2957]) ).
thf(10654,plain,
( ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[10653:[bind(A,$thf( subset_relation ))]]) ).
thf(10747,plain,
( ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(simp,[status(thm)],[10654]) ).
thf(148,axiom,
! [B: $i,A: $i] :
( ( ( A @ member_of )
= B )
| ~ ( A @ ( B @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_singletons1) ).
thf(500,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)],[148]) ).
thf(2144,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( null_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,1631]) ).
thf(2145,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2144]) ).
thf(2175,plain,
! [A: $i] :
( ( A != y )
| ( null_class != w )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2145]) ).
thf(2184,plain,
( ( null_class != w )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2175]) ).
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/sandbox2/benchmark/theBenchmark.p',corollary_1_to_number_of_elements_in_class) ).
thf(183,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(184,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)],[183]) ).
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(lifteq,[status(thm)],[184]) ).
thf(129,axiom,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',special_classes_lemma) ).
thf(462,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(463,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[462]) ).
thf(3186,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)],[511,463]) ).
thf(3187,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[3186]) ).
thf(3188,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3187:[bind(A,$thf( F @ ( F @ complement @ intersection ) )),bind(B,$thf( F @ ( F @ complement @ intersection ) @ regular )),bind(C,$thf( F ))]]) ).
thf(3378,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[3188]) ).
thf(8498,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ power_class )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3378,1718]) ).
thf(8499,plain,
( ( null_class @ domain_of @ power_class )
= ( identity_relation @ complement @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[8498:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(3997,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( B @ diagonalise )
= ( null_class @ domain_of @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3378,255]) ).
thf(3998,plain,
( ( identity_relation @ complement @ diagonalise )
= ( null_class @ domain_of @ complement ) ),
inference(pattern_uni,[status(thm)],[3997:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(8684,plain,
( ( null_class @ domain_of @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(rewrite,[status(thm)],[8499,3998]) ).
thf(8778,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[8684,194]) ).
thf(8779,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[8778:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(105,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(404,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(405,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[404]) ).
thf(4774,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[405,4700]) ).
thf(4775,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4774:[bind(A,$thf( choice ))]]) ).
thf(80,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip1) ).
thf(344,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(2192,plain,
! [B: $i,A: $i] :
( ( B != universal_class )
| ( A != y )
| ( null_class != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,2183]) ).
thf(2193,plain,
! [A: $i] :
( ( A != y )
| ( null_class != y )
| ~ ( A @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2192:[bind(A,$thf( A )),bind(B,$thf( universal_class ))]]) ).
thf(2194,plain,
( ( null_class != y )
| ~ ( y @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[2193]) ).
thf(11050,plain,
( ( null_class != y )
| ~ ( y @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[2194,482]) ).
thf(11051,plain,
( ( null_class != y )
| ~ ( y @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[11050]) ).
thf(12,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(195,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(196,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[195]) ).
thf(9,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(186,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(187,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[186]) ).
thf(188,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[187]) ).
thf(189,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[188]) ).
thf(4001,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)],[194,3378]) ).
thf(4002,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[4001:[bind(A,$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)],[4002]) ).
thf(8818,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[8684,4034]) ).
thf(8819,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[8818:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(54,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton @ ( A @ singleton @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_and_singletons) ).
thf(284,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton @ ( A @ singleton @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(110,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain1) ).
thf(416,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)],[110]) ).
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/sandbox2/benchmark/theBenchmark.p',ordered_pair_determines_components2) ).
thf(325,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(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/sandbox2/benchmark/theBenchmark.p',rotate2) ).
thf(289,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(124,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A = B )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality2) ).
thf(448,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)],[124]) ).
thf(69,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_non_singleton_unique1) ).
thf(316,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(3178,plain,
! [A: $i] :
( ( A
!= ( w @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,585]) ).
thf(3179,plain,
! [A: $i] :
( ( A
!= ( w @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3178]) ).
thf(3180,plain,
w @ singleton @ ( w @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3179:[bind(A,$thf( w @ singleton ))]]) ).
thf(626,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[257,463]) ).
thf(632,plain,
! [B: $i,A: $i] :
( ( B != subset_relation )
| ( ( B @ complement )
!= ( subset_relation @ inverse ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[626]) ).
thf(638,plain,
! [A: $i] :
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[632]) ).
thf(5,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
thf(178,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(179,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[178]) ).
thf(122,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/sandbox2/benchmark/theBenchmark.p',ordered_pair_determines_components1) ).
thf(443,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)],[122]) ).
thf(93,axiom,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_null_class) ).
thf(375,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(376,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[375]) ).
thf(141,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_12to_ordered_pairs_are_sets) ).
thf(485,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(9485,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( w @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[244,4155]) ).
thf(9486,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( w @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9485]) ).
thf(9539,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( w @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9486]) ).
thf(9549,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( w @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[9539]) ).
thf(10050,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( w @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[276,9549]) ).
thf(10051,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( w @ singleton ) ) ),
inference(simp,[status(thm)],[10050]) ).
thf(10085,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( w @ singleton ) ) ),
inference(simp,[status(thm)],[10051]) ).
thf(2924,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,276]) ).
thf(2925,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2924:[bind(A,$thf( omega ))]]) ).
thf(153,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair3) ).
thf(514,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(82,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair2) ).
thf(348,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(84,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product1) ).
thf(353,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)],[84]) ).
thf(6056,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,6021]) ).
thf(6057,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[6056]) ).
thf(6081,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( omega @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[6057]) ).
thf(20,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(210,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(211,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[210]) ).
thf(39,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(252,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(253,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(lifteq,[status(thm)],[252]) ).
thf(102,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product4) ).
thf(397,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)],[102]) ).
thf(3999,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)],[255,3378]) ).
thf(4000,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3999:[bind(A,$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)],[4000]) ).
thf(113,axiom,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_singletons_are_sets) ).
thf(423,plain,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(6187,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)],[423,376]) ).
thf(6188,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[6187]) ).
thf(6213,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[6188]) ).
thf(6221,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[6213]) ).
thf(116,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement2) ).
thf(428,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(45,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(265,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(3565,plain,
! [A: $i] :
( ( ( w @ singleton @ ( w @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3180,638]) ).
thf(3566,plain,
! [A: $i] :
( ( ( w @ singleton @ ( w @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3565]) ).
thf(3599,plain,
! [A: $i] :
( ( ( w @ singleton )
!= identity_relation )
| ( ( w @ singleton @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3566]) ).
thf(3606,plain,
( ( ( w @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3599]) ).
thf(17,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible2) ).
thf(204,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(205,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[204]) ).
thf(206,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ compatible ) ) )
| ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[205]) ).
thf(9460,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[276,244]) ).
thf(9461,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) ) ),
inference(simp,[status(thm)],[9460]) ).
thf(9462,plain,
omega @ ( omega @ ( successor_relation @ image ) @ subclass ),
inference(pattern_uni,[status(thm)],[9461:[bind(A,$thf( omega ))]]) ).
thf(9583,plain,
! [A: $i] :
( ( A != successor_relation )
| ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,9462]) ).
thf(9584,plain,
( ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[9583:[bind(A,$thf( successor_relation ))]]) ).
thf(1,negated_conjecture,
( ( x @ ( w @ ordered_pair ) )
= ( z @ ( y @ ordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_ordered_pair_determines_components1_1) ).
thf(169,plain,
( ( x @ ( w @ ordered_pair ) )
= ( z @ ( y @ ordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(172,plain,
( ( z @ ( y @ ordered_pair ) )
= ( x @ ( w @ ordered_pair ) ) ),
inference(lifteq,[status(thm)],[169]) ).
thf(112,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_is_set) ).
thf(422,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(660,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)],[422,321]) ).
thf(661,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)],[660]) ).
thf(662,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[661:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ ordered_pair ) ))]]) ).
thf(669,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[662]) ).
thf(733,plain,
! [B: $i,A: $i] :
( ( ( z @ ( y @ ordered_pair ) )
!= ( B @ ( A @ ordered_pair ) ) )
| ( ( x @ ( w @ ordered_pair ) @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[172,669]) ).
thf(734,plain,
( ( x @ ( w @ ordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[733:[bind(A,$thf( y )),bind(B,$thf( z ))]]) ).
thf(26,axiom,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair1) ).
thf(221,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(59,axiom,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_member_of_ordered_pair) ).
thf(294,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(2522,plain,
! [B: $i,A: $i] :
( ( ( z @ ( y @ ordered_pair ) )
!= ( B @ ( A @ ordered_pair ) ) )
| ( x @ ( w @ ordered_pair ) @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[172,294]) ).
thf(2523,plain,
x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ),
inference(pattern_uni,[status(thm)],[2522:[bind(A,$thf( y )),bind(B,$thf( z ))]]) ).
thf(126,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets3) ).
thf(453,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
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/sandbox2/benchmark/theBenchmark.p',corollary_1_to_unordered_pair) ).
thf(312,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(4,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists4) ).
thf(175,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(176,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(cnf,[status(esa)],[175]) ).
thf(177,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[176]) ).
thf(553,plain,
! [A: $i] :
( ( ( universal_class @ ( w @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[170,376]) ).
thf(554,plain,
! [A: $i] :
( ( universal_class @ ( w @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[553]) ).
thf(555,plain,
! [A: $i] :
( ( null_class != universal_class )
| ( w != A ) ),
inference(simp,[status(thm)],[554]) ).
thf(556,plain,
null_class != universal_class,
inference(simp,[status(thm)],[555]) ).
thf(3248,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,556]) ).
thf(3249,plain,
! [A: $i] :
( ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3248:[]]) ).
thf(3401,plain,
universal_class @ ( universal_class @ regular @ member ),
inference(simp,[status(thm)],[3249]) ).
thf(3441,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[3401,321]) ).
thf(3442,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3441]) ).
thf(3443,plain,
( ( universal_class @ regular @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[3442:[bind(A,$thf( universal_class @ regular ))]]) ).
thf(3490,plain,
! [A: $i] :
( ( ( A @ member_of @ singleton )
!= ( universal_class @ regular @ singleton ) )
| ( A != null_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[177,3443]) ).
thf(3492,plain,
( ( ( universal_class @ regular )
!= ( null_class @ member_of ) )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[3490]) ).
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/sandbox2/benchmark/theBenchmark.p',flip2) ).
thf(292,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(32,axiom,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',two_subsets_of_singleton) ).
thf(233,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(248,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[247]) ).
thf(95,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(380,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(152,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( A @ unordered_pair ) @ subclass ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_is_subset) ).
thf(512,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)],[152]) ).
thf(15,axiom,
! [A: $i] : ( A @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_is_reflexive) ).
thf(201,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(9587,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,9462]) ).
thf(9588,plain,
( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[9587:[bind(A,$thf( omega ))]]) ).
thf(25,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(219,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(220,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[219]) ).
thf(162,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality1) ).
thf(534,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)],[162]) ).
thf(1610,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ~ $true
| ~ ( w @ ( y @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[482,1259]) ).
thf(1611,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ~ ( w @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1610]) ).
thf(1627,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != w )
| ~ ( w @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1611]) ).
thf(1635,plain,
( ( universal_class != y )
| ~ ( w @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1627]) ).
thf(2645,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ( universal_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[291,1635]) ).
thf(2646,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[2645]) ).
thf(2660,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= w )
| ( element_relation != y )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[2646]) ).
thf(75,axiom,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singletons_are_sets) ).
thf(334,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(1722,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[405,257]) ).
thf(1723,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[1722:[bind(A,$thf( subset_relation ))]]) ).
thf(2559,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)],[2523,638]) ).
thf(2560,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[2559]) ).
thf(2593,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) )
!= identity_relation )
| ( ( y @ singleton )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[2560]) ).
thf(2596,plain,
( ( ( x @ ( w @ ordered_pair ) )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[2593]) ).
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/sandbox2/benchmark/theBenchmark.p',corollary_2_to_number_of_elements_in_class) ).
thf(302,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(43,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(260,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(2186,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( universal_class != y )
| ( B != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,2183]) ).
thf(2187,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != y )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2186:[bind(A,$thf( null_class ))]]) ).
thf(2195,plain,
( ( universal_class != y )
| ~ ( null_class @ ( y @ subclass ) )
| ~ ( y @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[2187]) ).
thf(11483,plain,
( ( universal_class != y )
| ~ ( null_class @ ( y @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[2195,216]) ).
thf(11484,plain,
( ( universal_class != y )
| ~ ( null_class @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[11483]) ).
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/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_sets1) ).
thf(297,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(114,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(424,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)],[114]) ).
thf(3469,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,3443]) ).
thf(3470,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3469]) ).
thf(3471,plain,
universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3470:[bind(A,$thf( universal_class @ regular @ singleton ))]]) ).
thf(1908,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( null_class != w )
| ( B != w )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,1892]) ).
thf(1909,plain,
! [A: $i] :
( ( null_class != w )
| ( A != w )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1908:[bind(A,$thf( universal_class ))]]) ).
thf(1916,plain,
( ( null_class != w )
| ~ ( universal_class @ ( w @ subclass ) )
| ~ ( w @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1909]) ).
thf(7016,plain,
( ( null_class != w )
| ~ $true
| ~ ( w @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[1916,482]) ).
thf(7017,plain,
( ( null_class != w )
| ~ ( w @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[7016]) ).
thf(7022,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != w )
| ~ ( w @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,7017]) ).
thf(7023,plain,
! [A: $i] :
( ( A != w )
| ~ ( w @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7022:[]]) ).
thf(7084,plain,
( ~ ( w @ ( universal_class @ subclass ) )
| ( w @ ( w @ regular @ member ) ) ),
inference(simp,[status(thm)],[7023]) ).
thf(2411,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ( null_class != w )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,1633]) ).
thf(2412,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ( null_class != w ) ),
inference(simp,[status(thm)],[2411]) ).
thf(2445,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= w )
| ( successor_relation != y )
| ( null_class != w ) ),
inference(simp,[status(thm)],[2412]) ).
thf(13,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class1) ).
thf(197,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(9585,plain,
! [A: $i] :
( ( A != omega )
| ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,9462]) ).
thf(9586,plain,
( ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[9585:[bind(A,$thf( omega ))]]) ).
thf(156,axiom,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists1) ).
thf(522,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(145,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/sandbox2/benchmark/theBenchmark.p',property_1_of_ordered_pair) ).
thf(493,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)],[145]) ).
thf(127,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_3) ).
thf(456,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)],[127]) ).
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/sandbox2/benchmark/theBenchmark.p',symmetric_difference) ).
thf(229,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(119,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality4) ).
thf(435,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)],[119]) ).
thf(6242,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)],[179,6221]) ).
thf(6243,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[6242:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(6260,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[6243]) ).
thf(6431,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ ( D @ singleton @ unordered_pair ) ) )
| ( ( B @ ( A @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[179,6260]) ).
thf(6432,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[6431:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( A )),bind(D,$thf( E ))]]) ).
thf(6445,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[6432]) ).
thf(155,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_singletons_are_sets) ).
thf(519,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(164,axiom,
null_class @ singleton @ ( null_class @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_in_its_singleton) ).
thf(540,plain,
null_class @ singleton @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(566,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[540,376]) ).
thf(567,plain,
! [A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[566]) ).
thf(569,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= null_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[567]) ).
thf(571,plain,
( ( null_class @ singleton )
!= null_class ),
inference(simp,[status(thm)],[569]) ).
thf(1097,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,571]) ).
thf(1098,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( null_class @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1097:[bind(A,$thf( null_class @ singleton )),bind(B,$thf( B ))]]) ).
thf(1195,plain,
( ~ ( null_class @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[1098]) ).
thf(5438,plain,
( ~ $true
| ~ ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1195,216]) ).
thf(5439,plain,
~ ( null_class @ ( null_class @ singleton @ subclass ) ),
inference(simp,[status(thm)],[5438]) ).
thf(5476,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,5439]) ).
thf(5477,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[5476]) ).
thf(5502,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( null_class @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[5477]) ).
thf(724,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)],[179,655]) ).
thf(725,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[724:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(1117,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,556]) ).
thf(1118,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1117:[bind(A,$thf( null_class ))]]) ).
thf(1202,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1118]) ).
thf(1452,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[1202,482]) ).
thf(1453,plain,
~ ( null_class @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[1452]) ).
thf(9505,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[244,1453]) ).
thf(9506,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9505]) ).
thf(9535,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9506]) ).
thf(9576,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[9535]) ).
thf(98,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists3) ).
thf(388,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
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/sandbox2/benchmark/theBenchmark.p',domain) ).
thf(225,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(78,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(339,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(9635,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9462,1453]) ).
thf(9636,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[9635]) ).
thf(9662,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[9636]) ).
thf(158,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/sandbox2/benchmark/theBenchmark.p',corollary_1_to_cartesian_product) ).
thf(526,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)],[158]) ).
thf(143,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function3) ).
thf(489,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)],[143]) ).
thf(167,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_singleton_is_unique) ).
thf(547,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(21,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(212,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
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/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).
thf(328,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(133,axiom,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( B @ singleton @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_member_of_ordered_pair) ).
thf(471,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( B @ singleton @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(163,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom1) ).
thf(537,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(100,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair) ).
thf(393,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)],[100]) ).
thf(2565,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( y @ singleton ) )
| ( x @ ( w @ ordered_pair ) @ ( null_class @ member ) )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[411,2523]) ).
thf(2566,plain,
( ( x @ ( w @ ordered_pair ) @ ( null_class @ member ) )
| ( universal_class @ ( y @ member ) ) ),
inference(pattern_uni,[status(thm)],[2565:[bind(A,$thf( y ))]]) ).
thf(213,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[212]) ).
thf(4347,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(rewrite,[status(thm)],[213,181]) ).
thf(597,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ~ $true
| ( ( B @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[334,321]) ).
thf(598,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( ( B @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[597]) ).
thf(599,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[598:[bind(A,$thf( C )),bind(B,$thf( C @ singleton ))]]) ).
thf(605,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(simp,[status(thm)],[599]) ).
thf(3084,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,605]) ).
thf(3085,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3084]) ).
thf(3086,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[3085:[bind(A,$thf( D @ singleton @ singleton )),bind(B,$thf( D ))]]) ).
thf(3333,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) ),
inference(simp,[status(thm)],[3086]) ).
thf(149,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation2) ).
thf(503,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(3028,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,734]) ).
thf(3029,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3028]) ).
thf(3030,plain,
x @ ( w @ ordered_pair ) @ singleton @ ( x @ ( w @ ordered_pair ) @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3029:[bind(A,$thf( x @ ( w @ ordered_pair ) @ singleton ))]]) ).
thf(4513,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)],[3030,463]) ).
thf(4514,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)],[4513]) ).
thf(4520,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)],[4514]) ).
thf(4535,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( x @ ( w @ ordered_pair ) @ singleton ) ),
inference(simp,[status(thm)],[4520]) ).
thf(2866,plain,
! [A: $i] :
( ( null_class != null_class )
| ( null_class != w )
| ( A != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,2184]) ).
thf(2867,plain,
! [A: $i] :
( ( null_class != w )
| ( A != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2866:[]]) ).
thf(3336,plain,
( ( null_class != w )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[2867]) ).
thf(3864,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != w )
| ( y @ ( y @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,3336]) ).
thf(3865,plain,
! [A: $i] :
( ( A != w )
| ( y @ ( y @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3864:[]]) ).
thf(3903,plain,
( ( y @ ( y @ regular @ member ) )
| ( w @ ( w @ regular @ member ) ) ),
inference(simp,[status(thm)],[3865]) ).
thf(266,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[265]) ).
thf(618,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[540,463]) ).
thf(619,plain,
! [B: $i,A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[618]) ).
thf(627,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( null_class @ singleton ) )
| ( null_class != A ) ),
inference(simp,[status(thm)],[619]) ).
thf(633,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( null_class @ singleton ) ),
inference(simp,[status(thm)],[627]) ).
thf(165,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_unordered_pair) ).
thf(541,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)],[165]) ).
thf(160,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(530,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(108,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/sandbox2/benchmark/theBenchmark.p',flip3) ).
thf(412,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)],[108]) ).
thf(117,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_equals_singleton2) ).
thf(430,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(3595,plain,
! [B: $i,A: $i] :
( ( ( w @ singleton @ ( w @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3180,463]) ).
thf(3596,plain,
! [B: $i,A: $i] :
( ( w @ singleton @ ( w @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3595]) ).
thf(3601,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( w @ singleton ) )
| ( ( w @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3596]) ).
thf(3607,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( w @ singleton ) ),
inference(simp,[status(thm)],[3601]) ).
thf(44,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(263,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(264,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[263]) ).
thf(52,axiom,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_of_null_class_is_subclass) ).
thf(279,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
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/sandbox2/benchmark/theBenchmark.p',homomorphism4) ).
thf(286,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(9615,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( w @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9462,4155]) ).
thf(9616,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( w @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[9615]) ).
thf(9663,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( w @ singleton ) ) ),
inference(simp,[status(thm)],[9616]) ).
thf(42,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cantor_class) ).
thf(258,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(259,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[258]) ).
thf(2958,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,257]) ).
thf(2959,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2958:[bind(A,$thf( subset_relation ))]]) ).
thf(624,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( w @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[170,463]) ).
thf(625,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( w @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[624]) ).
thf(629,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= universal_class )
| ( w != A ) ),
inference(simp,[status(thm)],[625]) ).
thf(635,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[629]) ).
thf(639,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != universal_class ) ),
inference(paramod_ordered,[status(thm)],[257,635]) ).
thf(640,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( A @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[639]) ).
thf(641,plain,
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[640]) ).
thf(99,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/sandbox2/benchmark/theBenchmark.p',element_relation3) ).
thf(391,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)],[99]) ).
thf(96,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists2) ).
thf(382,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(147,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/sandbox2/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
thf(498,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)],[147]) ).
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/sandbox2/benchmark/theBenchmark.p',number_of_elements_in_class) ).
thf(335,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(9497,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[244,5439]) ).
thf(9498,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9497]) ).
thf(9525,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9498]) ).
thf(9563,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[9525]) ).
thf(10561,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[276,9563]) ).
thf(10562,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[10561]) ).
thf(10596,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[10562]) ).
thf(3089,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,579]) ).
thf(3090,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3089]) ).
thf(3091,plain,
omega @ singleton @ ( omega @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3090:[bind(A,$thf( omega @ singleton ))]]) ).
thf(83,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets4) ).
thf(350,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(5215,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ ( B @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[3378,4033]) ).
thf(5216,plain,
( ( null_class @ domain_of @ ( identity_relation @ complement @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[5215:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(5314,plain,
( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
= null_class ),
inference(rewrite,[status(thm)],[5216,3998]) ).
thf(131,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/sandbox2/benchmark/theBenchmark.p',cartesian_product3) ).
thf(466,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)],[131]) ).
thf(9807,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class )
!= ( A @ power_class ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[8779,194]) ).
thf(9808,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[9807:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(33,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(236,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(106,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom2) ).
thf(406,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(49,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(274,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(111,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
!= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_ordered_pairs_are_sets) ).
thf(419,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(6177,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)],[196,423]) ).
thf(6178,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[6177:[bind(A,$thf( D @ singleton )),bind(B,$thf( D )),bind(C,$thf( D @ singleton ))]]) ).
thf(6217,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[6178]) ).
thf(94,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets1) ).
thf(377,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)],[94]) ).
thf(230,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ symmetric_difference ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) ) ),
inference(lifteq,[status(thm)],[229]) ).
thf(36,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',transitivity_of_subclass) ).
thf(245,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(87,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_equals_singleton1) ).
thf(362,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(125,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection3) ).
thf(451,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)],[125]) ).
thf(48,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_non_singleton_unique2) ).
thf(271,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
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/sandbox2/benchmark/theBenchmark.p',compatible3) ).
thf(208,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(2433,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ( null_class != w )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[291,1633]) ).
thf(2434,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( w @ ( y @ subclass ) ) )
| ( null_class != w ) ),
inference(simp,[status(thm)],[2433]) ).
thf(2446,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= w )
| ( element_relation != y )
| ( null_class != w ) ),
inference(simp,[status(thm)],[2434]) ).
thf(3954,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)],[3471,463]) ).
thf(3955,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)],[3954]) ).
thf(3958,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) )
| ( ( universal_class @ regular @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3955]) ).
thf(3971,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) ),
inference(simp,[status(thm)],[3958]) ).
thf(198,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[197]) ).
thf(136,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(477,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(4187,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( w @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[291,4155]) ).
thf(4188,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( w @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[4187]) ).
thf(4203,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( w @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[4188]) ).
thf(81,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(345,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(89,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/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_sets2) ).
thf(367,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)],[89]) ).
thf(2563,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2523,376]) ).
thf(2564,plain,
! [A: $i] :
( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2563]) ).
thf(2594,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) )
!= null_class )
| ( ( y @ singleton )
!= A ) ),
inference(simp,[status(thm)],[2564]) ).
thf(2597,plain,
( ( x @ ( w @ ordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2594]) ).
thf(3163,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,2597]) ).
thf(3164,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3163]) ).
thf(3165,plain,
x @ ( w @ ordered_pair ) @ ( x @ ( w @ ordered_pair ) @ regular @ member ),
inference(pattern_uni,[status(thm)],[3164:[bind(A,$thf( x @ ( w @ ordered_pair ) ))]]) ).
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/sandbox2/benchmark/theBenchmark.p',compatible4) ).
thf(240,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(250,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[249]) ).
thf(251,plain,
! [A: $i] :
( ~ ( A @ operation )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[250]) ).
thf(2528,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[294,376]) ).
thf(2529,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2528]) ).
thf(2546,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[2529]) ).
thf(2550,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2546]) ).
thf(1400,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)],[194,194]) ).
thf(1401,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1400:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1449,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1401]) ).
thf(8820,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ power_class )
!= ( A @ power_class ) )
| ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[8684,1449]) ).
thf(8821,plain,
( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[8820:[bind(A,$thf( null_class @ domain_of ))]]) ).
thf(1878,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( universal_class != w )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[482,1630]) ).
thf(1879,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1878]) ).
thf(1880,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != w )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1879]) ).
thf(1891,plain,
( ( universal_class != y )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1880]) ).
thf(1896,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( universal_class != y )
| ( B != w )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,1891]) ).
thf(1897,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != w )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1896:[bind(A,$thf( universal_class ))]]) ).
thf(1904,plain,
( ( universal_class != y )
| ~ ( universal_class @ ( w @ subclass ) )
| ~ ( w @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1897]) ).
thf(6782,plain,
( ( universal_class != y )
| ~ $true
| ~ ( w @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[1904,482]) ).
thf(6783,plain,
( ( universal_class != y )
| ~ ( w @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[6782]) ).
thf(1454,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,1453]) ).
thf(1455,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1454]) ).
thf(1477,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[1455]) ).
thf(46,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(267,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(120,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/sandbox2/benchmark/theBenchmark.p',homomorphism5) ).
thf(438,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)],[120]) ).
thf(8824,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[8684,1449]) ).
thf(8825,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[8824:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(12287,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
!= ( A @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[8825,1449]) ).
thf(12288,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[12287:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(142,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(487,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(154,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/sandbox2/benchmark/theBenchmark.p',right_cancellation) ).
thf(516,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)],[154]) ).
thf(1898,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( B != y )
| ( universal_class != w )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,1891]) ).
thf(1899,plain,
! [A: $i] :
( ( A != y )
| ( universal_class != w )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1898:[bind(A,$thf( universal_class ))]]) ).
thf(1905,plain,
( ( universal_class != w )
| ~ ( universal_class @ ( y @ subclass ) )
| ~ ( y @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1899]) ).
thf(6852,plain,
( ( universal_class != w )
| ~ $true
| ~ ( y @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[1905,482]) ).
thf(6853,plain,
( ( universal_class != w )
| ~ ( y @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[6852]) ).
thf(85,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/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom3) ).
thf(355,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)],[85]) ).
thf(60,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_singletons2) ).
thf(295,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(268,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[267]) ).
thf(14660,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[268,1453]) ).
thf(14661,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14660]) ).
thf(14694,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14661]) ).
thf(14727,plain,
( ( omega != universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14694]) ).
thf(14757,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[276,14727]) ).
thf(14758,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[14757]) ).
thf(14775,plain,
( ( omega != null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[14758]) ).
thf(53,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one2) ).
thf(282,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(241,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)],[240]) ).
thf(242,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)],[241]) ).
thf(9061,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)],[242,181]) ).
thf(3207,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,571]) ).
thf(3208,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3207]) ).
thf(3209,plain,
null_class @ singleton @ ( null_class @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3208:[bind(A,$thf( null_class @ singleton ))]]) ).
thf(66,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(310,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(14,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one3) ).
thf(199,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
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/sandbox2/benchmark/theBenchmark.p',property_3_of_ordered_pair) ).
thf(331,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(3538,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3091,463]) ).
thf(3539,plain,
! [B: $i,A: $i] :
( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3538]) ).
thf(3544,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( omega @ singleton ) )
| ( ( omega @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3539]) ).
thf(3554,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[3544]) ).
thf(77,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate1) ).
thf(338,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(226,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)],[225]) ).
thf(6159,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)],[226,203]) ).
thf(166,axiom,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_unique) ).
thf(544,plain,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(90,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product2) ).
thf(370,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)],[90]) ).
thf(741,plain,
! [A: $i] :
( ( ( A @ member_of @ singleton )
!= ( x @ ( w @ ordered_pair ) @ singleton ) )
| ( A != null_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[177,734]) ).
thf(744,plain,
( ( ( x @ ( w @ ordered_pair ) )
!= ( null_class @ member_of ) )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[741]) ).
thf(9627,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9462,5439]) ).
thf(9628,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[9627]) ).
thf(9675,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[9628]) ).
thf(4312,plain,
! [A: $i] :
( ( ( w @ ( w @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class )
| ( universal_class != w ) ),
inference(paramod_ordered,[status(thm)],[3399,321]) ).
thf(4313,plain,
! [A: $i] :
( ( ( w @ ( w @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[4312]) ).
thf(4335,plain,
! [A: $i] :
( ( universal_class != w )
| ( ( w @ regular )
!= A )
| ( ( A @ singleton )
!= null_class )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[4313]) ).
thf(4342,plain,
( ( ( w @ regular @ singleton )
!= null_class )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[4335]) ).
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/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_1) ).
thf(314,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(594,plain,
! [A: $i] :
( ( ( A @ member_of @ singleton )
!= ( w @ singleton ) )
| ( A != null_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[177,585]) ).
thf(595,plain,
( ( ( null_class @ member_of @ singleton )
!= ( w @ singleton ) )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[594]) ).
thf(1854,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( universal_class != w )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,1630]) ).
thf(1855,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1854]) ).
thf(1881,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= y )
| ( successor_relation != w )
| ( universal_class != w ) ),
inference(simp,[status(thm)],[1855]) ).
thf(146,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/sandbox2/benchmark/theBenchmark.p',subset_relation) ).
thf(496,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)],[146]) ).
thf(497,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)],[496]) ).
thf(123,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_in_its_singleton) ).
thf(446,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(4546,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( w @ regular @ singleton ) )
| ( null_class != null_class )
| ( universal_class != w )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[411,4342]) ).
thf(4547,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( w @ regular @ singleton ) )
| ( universal_class != w )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4546]) ).
thf(4548,plain,
( ( universal_class != w )
| ( universal_class @ ( w @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4547:[bind(A,$thf( w @ regular ))]]) ).
thf(5149,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)],[3333,463]) ).
thf(5150,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)],[5149]) ).
thf(5161,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton )
!= ( C @ ( C @ complement @ intersection ) ) )
| ( ( A @ singleton @ singleton @ regular )
!= B ) ),
inference(simp,[status(thm)],[5150]) ).
thf(5179,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ singleton )
!= ( B @ ( B @ complement @ intersection ) ) ),
inference(simp,[status(thm)],[5161]) ).
thf(9931,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[276,9576]) ).
thf(9932,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[9931]) ).
thf(9966,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[9932]) ).
thf(157,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',replacement) ).
thf(524,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)],[157]) ).
thf(218,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[217]) ).
thf(1910,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != w )
| ( universal_class != w )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,1892]) ).
thf(1911,plain,
! [A: $i] :
( ( A != w )
| ( universal_class != w )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1910:[bind(A,$thf( null_class ))]]) ).
thf(1917,plain,
( ( universal_class != w )
| ~ ( null_class @ ( w @ subclass ) )
| ~ ( w @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1911]) ).
thf(8881,plain,
( ( universal_class != w )
| ~ ( null_class @ ( w @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[1917,216]) ).
thf(8882,plain,
( ( universal_class != w )
| ~ ( null_class @ ( w @ subclass ) ) ),
inference(simp,[status(thm)],[8881]) ).
thf(104,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/sandbox2/benchmark/theBenchmark.p',domain2) ).
thf(401,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)],[104]) ).
thf(5480,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[291,5439]) ).
thf(5481,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[5480]) ).
thf(5505,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( null_class @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[5481]) ).
thf(18,axiom,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair2) ).
thf(207,plain,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
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/sandbox2/benchmark/theBenchmark.p',range) ).
thf(227,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(228,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)],[227]) ).
thf(6412,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)],[228,203]) ).
thf(9589,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9462,6021]) ).
thf(9590,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[9589]) ).
thf(9660,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[9590]) ).
thf(3508,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)],[3091,638]) ).
thf(3509,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3508]) ).
thf(3542,plain,
! [A: $i] :
( ( ( omega @ singleton )
!= identity_relation )
| ( ( omega @ singleton @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3509]) ).
thf(3552,plain,
( ( ( omega @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3542]) ).
thf(234,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[233]) ).
thf(235,plain,
! [B: $i,A: $i] :
( ~ ( B @ singleton @ ( A @ subclass ) )
| ( ( B @ singleton )
= A )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[234]) ).
thf(246,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[245]) ).
thf(209,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[208]) ).
thf(3497,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)],[209,181]) ).
thf(118,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_members) ).
thf(433,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(2889,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != y )
| ~ ( y @ ( w @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,1631]) ).
thf(2890,plain,
! [A: $i] :
( ( A != y )
| ~ ( y @ ( w @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2889:[]]) ).
thf(3351,plain,
( ~ ( y @ ( w @ subclass ) )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[2890]) ).
thf(261,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[260]) ).
thf(262,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[261]) ).
thf(12116,plain,
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( 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(paramod_ordered,[status(thm)],[8779,8825]) ).
thf(12117,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)],[12116:[]]) ).
thf(2587,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2523,463]) ).
thf(2588,plain,
! [B: $i,A: $i] :
( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2587]) ).
thf(2591,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( x @ ( w @ ordered_pair ) ) )
| ( ( y @ singleton )
!= A ) ),
inference(simp,[status(thm)],[2588]) ).
thf(2604,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( x @ ( w @ ordered_pair ) ) ),
inference(simp,[status(thm)],[2591]) ).
thf(1470,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[291,1453]) ).
thf(1471,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1470]) ).
thf(1480,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( element_relation != universal_class ) ),
inference(simp,[status(thm)],[1471]) ).
thf(2280,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != w )
| ( null_class != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,2184]) ).
thf(2281,plain,
! [A: $i] :
( ( A != w )
| ( null_class != y )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2280:[bind(A,$thf( null_class ))]]) ).
thf(2287,plain,
( ( null_class != y )
| ~ ( null_class @ ( w @ subclass ) )
| ~ ( w @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[2281]) ).
thf(13095,plain,
( ( null_class != y )
| ~ ( null_class @ ( w @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[2287,216]) ).
thf(13096,plain,
( ( null_class != y )
| ~ ( null_class @ ( w @ subclass ) ) ),
inference(simp,[status(thm)],[13095]) ).
thf(92,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/sandbox2/benchmark/theBenchmark.p',compose3) ).
thf(373,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)],[92]) ).
thf(2278,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( null_class != w )
| ( B != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[192,2184]) ).
thf(2279,plain,
! [A: $i] :
( ( null_class != w )
| ( A != y )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2278:[bind(A,$thf( null_class ))]]) ).
thf(2286,plain,
( ( null_class != w )
| ~ ( null_class @ ( y @ subclass ) )
| ~ ( y @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[2279]) ).
thf(12390,plain,
( ( null_class != w )
| ~ ( null_class @ ( y @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[2286,216]) ).
thf(12391,plain,
( ( null_class != w )
| ~ ( null_class @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[12390]) ).
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/sandbox2/benchmark/theBenchmark.p',compose2) ).
thf(300,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(2868,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != w )
| ( null_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,2184]) ).
thf(2869,plain,
! [A: $i] :
( ( A != w )
| ( null_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2868:[]]) ).
thf(3337,plain,
( ( null_class != y )
| ( w @ ( w @ regular @ member ) ) ),
inference(simp,[status(thm)],[2869]) ).
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/sandbox2/benchmark/theBenchmark.p',left_cancellation) ).
thf(305,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(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/sandbox2/benchmark/theBenchmark.p',equality3) ).
thf(341,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(86,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/sandbox2/benchmark/theBenchmark.p',successor_relation3) ).
thf(358,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)],[86]) ).
thf(135,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets2) ).
thf(474,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)],[135]) ).
thf(103,axiom,
universal_class @ ( null_class @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_a_set) ).
thf(400,plain,
universal_class @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(161,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(532,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(682,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)],[540,638]) ).
thf(683,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[682]) ).
thf(691,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= identity_relation )
| ( null_class != A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[683]) ).
thf(696,plain,
( ( ( null_class @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[691]) ).
thf(150,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_identified_by_element1) ).
thf(506,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(12399,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != w )
| ~ ( null_class @ ( y @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,12391]) ).
thf(12400,plain,
! [A: $i] :
( ( A != w )
| ~ ( null_class @ ( y @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[12399:[]]) ).
thf(12476,plain,
( ~ ( null_class @ ( y @ subclass ) )
| ( w @ ( w @ regular @ member ) ) ),
inference(simp,[status(thm)],[12400]) ).
thf(101,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/sandbox2/benchmark/theBenchmark.p',rotate3) ).
thf(395,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)],[101]) ).
thf(115,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(427,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(596,plain,
( ( ( null_class @ member_of )
!= w )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[594]) ).
thf(159,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation2) ).
thf(528,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(128,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_5) ).
thf(459,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(200,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[199]) ).
thf(51,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction1) ).
thf(277,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(6181,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)],[179,423]) ).
thf(6182,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[6181:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(6222,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(simp,[status(thm)],[6182]) ).
thf(11056,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != y )
| ~ ( y @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,11051]) ).
thf(11057,plain,
! [A: $i] :
( ( A != y )
| ~ ( y @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[11056:[]]) ).
thf(11135,plain,
( ~ ( y @ ( universal_class @ subclass ) )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[11057]) ).
thf(2909,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != w )
| ~ ( w @ ( y @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,1633]) ).
thf(2910,plain,
! [A: $i] :
( ( A != w )
| ~ ( w @ ( y @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2909:[]]) ).
thf(3362,plain,
( ~ ( w @ ( y @ subclass ) )
| ( w @ ( w @ regular @ member ) ) ),
inference(simp,[status(thm)],[2910]) ).
thf(130,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(464,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
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/sandbox2/benchmark/theBenchmark.p',homomorphism6) ).
thf(322,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(4772,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4700,1453]) ).
thf(4773,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[4772]) ).
thf(4779,plain,
( ( identity_relation != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= universal_class ) ),
inference(simp,[status(thm)],[4773]) ).
thf(2158,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( null_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[291,1631]) ).
thf(2159,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2158]) ).
thf(2177,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= y )
| ( element_relation != w )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2159]) ).
thf(2136,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( null_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,1631]) ).
thf(2137,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( y @ ( w @ subclass ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2136]) ).
thf(2170,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= y )
| ( successor_relation != w )
| ( null_class != y ) ),
inference(simp,[status(thm)],[2137]) ).
thf(144,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(491,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(13104,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != y )
| ~ ( null_class @ ( w @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,13096]) ).
thf(13105,plain,
! [A: $i] :
( ( A != y )
| ~ ( null_class @ ( w @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[13104:[]]) ).
thf(13182,plain,
( ~ ( null_class @ ( w @ subclass ) )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[13105]) ).
thf(42368,plain,
$false,
inference(e,[status(thm)],[5810,6085,4201,1718,365,10638,1633,3399,479,249,2463,468,170,308,4960,247,269,202,1882,217,276,440,655,511,385,6939,550,472,10747,500,2184,185,8779,4775,174,404,2957,257,372,344,11051,196,1892,189,8819,284,416,325,289,448,316,3180,3998,638,179,443,321,376,485,10085,2925,514,348,353,6081,216,211,253,397,4033,6221,6021,411,243,428,265,3606,206,9584,3378,734,221,2523,453,312,3492,4700,292,233,248,380,512,201,9588,220,534,2660,334,9549,1723,2596,302,260,11484,297,424,3471,192,7084,2445,252,197,9586,522,493,456,229,484,4155,435,6445,519,1259,5502,725,9576,388,169,4653,225,339,9662,669,526,489,547,212,328,471,256,193,537,393,2566,4347,3409,3333,503,605,4535,3903,244,266,633,541,6260,530,412,430,3607,264,279,180,1631,375,423,286,8684,9663,204,259,2959,579,291,641,391,382,498,1635,335,181,10596,3091,350,540,5314,466,9808,236,406,172,219,274,419,6217,377,230,245,509,362,451,271,208,1630,5439,2446,3971,198,462,477,4203,345,367,3165,483,240,251,2550,8821,6783,1477,267,438,12288,9563,255,487,516,6853,194,2597,355,295,14775,282,414,9061,3209,409,263,310,9462,199,331,3554,463,338,6159,544,177,182,585,370,744,9675,175,4342,314,595,1881,497,446,571,231,4548,3401,5179,9966,556,524,8825,218,1453,482,203,319,8882,258,3336,186,401,171,5505,207,6412,2183,9660,3552,214,235,246,3497,433,7017,3351,262,12117,2604,405,190,422,1480,13096,373,12391,300,183,3337,305,210,294,341,358,195,474,400,635,532,696,9564,506,268,1891,12476,3030,4034,395,427,596,528,459,200,178,277,6222,11135,14727,3362,464,322,4779,215,3443,1449,222,232,2177,254,227,2170,491,13182,496]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET016-7 : TPTP v9.3.1. Bugfixed v7.3.0.
% 0.00/0.13 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.24/0.50 % Computer : n008.cluster.edu
% 0.24/0.50 % Model : x86_64 x86_64
% 0.24/0.50 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.24/0.50 % Memory : 8046.5625MB
% 0.24/0.50 % OS : Linux 6.8.0-71-generic
% 0.24/0.50 % CPULimit : 300
% 0.24/0.50 % WCLimit : 300
% 0.24/0.50 % DateTime : Sat Sep 26 06:51:55 UTC 2026
% 0.24/0.51 % CPUTime :
% 0.24/0.51 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.59/1.40 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.67/1.87 % [INFO] Parsing done (465ms).
% 2.67/1.89 % [INFO] Running in sequential loop mode.
% 4.18/2.50 % [INFO] eprover registered as external prover.
% 4.41/2.51 % [INFO] Scanning for conjecture ...
% 4.86/2.72 % [INFO] Found a conjecture (or negated_conjecture) and 165 axioms. Running axiom selection ...
% 5.35/2.91 % [INFO] Axiom selection finished. Selected 165 axioms (removed 0 axioms).
% 5.91/3.14 % [INFO] Problem is propositional (TPTP CNF).
% 6.21/3.17 % [INFO] Type checking passed.
% 6.21/3.18 % [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 ...
% 141.90/31.90 % External prover 'e' found a proof!
% 141.90/31.90 % [INFO] Killing All external provers ...
% 141.90/31.91 % Time passed: 31174ms (effective reasoning time: 30006ms)
% 141.90/31.91 % Axioms used in derivation (165): 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, 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, 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
% 141.90/31.91 % No. of inferences in proof: 856
% 141.90/31.91 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 31174 ms resp. 30006 ms w/o parsing
% 132.27/32.12 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 132.27/32.12 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------