%------------------------------------------------------------------------------
% File : LEO-II---2.3.1
% Problem : SET090-7 : TPTP v9.3.1. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp:raw
% Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n017.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 : Wed Oct 7 10:39:30 AM UTC 2026
% Result : Unsatisfiable 7.73s 1.53s
% Output : CNFRefutation 7.73s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 136
% Syntax : Number of formulae : 970 ( 583 unt; 0 typ; 0 def)
% Number of atoms : 4623 (1746 equ; 0 cnn)
% Maximal formula atoms : 5 ( 4 avg)
% Number of connectives : 9531 ( 898 ~;1219 |; 0 &;7414 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 49 usr; 11 con; 0-3 aty)
% Number of variables : 2156 ( 0 ^;2156 !; 0 ?;2156 :)
% Comments :
%------------------------------------------------------------------------------
thf(tp_apply,type,
apply: $i > $i > $i ).
thf(tp_cantor,type,
cantor: $i > $i ).
thf(tp_choice,type,
choice: $i ).
thf(tp_compatible,type,
compatible: $i > $i > $i > $o ).
thf(tp_complement,type,
complement: $i > $i ).
thf(tp_compose,type,
compose: $i > $i > $i ).
thf(tp_cross_product,type,
cross_product: $i > $i > $i ).
thf(tp_diagonalise,type,
diagonalise: $i > $i ).
thf(tp_domain,type,
domain: $i > $i > $i > $i ).
thf(tp_domain_of,type,
domain_of: $i > $i ).
thf(tp_element_relation,type,
element_relation: $i ).
thf(tp_first,type,
first: $i > $i ).
thf(tp_flip,type,
flip: $i > $i ).
thf(tp_function,type,
function: $i > $o ).
thf(tp_homomorphism,type,
homomorphism: $i > $i > $i > $o ).
thf(tp_identity_relation,type,
identity_relation: $i ).
thf(tp_image,type,
image: $i > $i > $i ).
thf(tp_inductive,type,
inductive: $i > $o ).
thf(tp_intersection,type,
intersection: $i > $i > $i ).
thf(tp_inverse,type,
inverse: $i > $i ).
thf(tp_member,type,
member: $i > $i > $o ).
thf(tp_member_of,type,
member_of: $i > $i ).
thf(tp_not_homomorphism1,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(tp_not_homomorphism2,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(tp_not_subclass_element,type,
not_subclass_element: $i > $i > $i ).
thf(tp_null_class,type,
null_class: $i ).
thf(tp_omega,type,
omega: $i ).
thf(tp_one_to_one,type,
one_to_one: $i > $o ).
thf(tp_operation,type,
operation: $i > $o ).
thf(tp_ordered_pair,type,
ordered_pair: $i > $i > $i ).
thf(tp_power_class,type,
power_class: $i > $i ).
thf(tp_range,type,
range: $i > $i > $i > $i ).
thf(tp_range_of,type,
range_of: $i > $i ).
thf(tp_regular,type,
regular: $i > $i ).
thf(tp_restrict,type,
restrict: $i > $i > $i > $i ).
thf(tp_rotate,type,
rotate: $i > $i ).
thf(tp_second,type,
second: $i > $i ).
thf(tp_single_valued_class,type,
single_valued_class: $i > $o ).
thf(tp_singleton,type,
singleton: $i > $i ).
thf(tp_subclass,type,
subclass: $i > $i > $o ).
thf(tp_subset_relation,type,
subset_relation: $i ).
thf(tp_successor,type,
successor: $i > $i ).
thf(tp_successor_relation,type,
successor_relation: $i ).
thf(tp_sum_class,type,
sum_class: $i > $i ).
thf(tp_symmetric_difference,type,
symmetric_difference: $i > $i > $i ).
thf(tp_u,type,
u: $i ).
thf(tp_union,type,
union: $i > $i > $i ).
thf(tp_universal_class,type,
universal_class: $i ).
thf(tp_unordered_pair,type,
unordered_pair: $i > $i > $i ).
thf(1,axiom,
! [X: $i] :
( ( ( singleton @ ( member_of @ X ) )
= X )
| ( ( member_of @ X )
= X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists4) ).
thf(2,axiom,
! [X: $i] :
( ( member @ ( member_of @ X ) @ universal_class )
| ( ( member_of @ X )
= X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists3) ).
thf(3,axiom,
! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( ( singleton @ ( member_of @ ( singleton @ Y ) ) )
= ( singleton @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists2) ).
thf(4,axiom,
! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( member @ ( member_of @ ( singleton @ Y ) ) @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists1) ).
thf(5,axiom,
! [Y: $i,Z: $i,X: $i] :
( ( ( unordered_pair @ Y @ Z )
!= ( singleton @ X ) )
| ~ ( member @ X @ universal_class )
| ( X = Y )
| ( X = Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).
thf(6,axiom,
! [X: $i,Y: $i] :
( ( ( singleton @ X )
!= ( singleton @ Y ) )
| ~ ( member @ Y @ universal_class )
| ( X = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_identified_by_element2) ).
thf(7,axiom,
! [X: $i,Y: $i] :
( ( ( singleton @ X )
!= ( singleton @ Y ) )
| ~ ( member @ X @ universal_class )
| ( X = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_identified_by_element1) ).
thf(8,axiom,
! [X: $i] :
( ( member @ X @ universal_class )
| ( ( singleton @ X )
= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_is_null_class) ).
thf(9,axiom,
! [Y: $i,X: $i] :
( ~ ( member @ Y @ ( singleton @ X ) )
| ( Y = X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',only_member_in_singleton) ).
thf(10,axiom,
member @ null_class @ ( singleton @ null_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_in_its_singleton) ).
thf(11,axiom,
! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( ( singleton @ X )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_set_in_its_singleton) ).
thf(12,axiom,
! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ X @ ( singleton @ X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_in_its_singleton) ).
thf(13,axiom,
! [Y: $i,X: $i] : ( member @ ( singleton @ Y ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_singletons_are_sets) ).
thf(14,axiom,
! [X: $i] : ( member @ ( singleton @ X ) @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singletons_are_sets) ).
thf(15,axiom,
! [X: $i,Z: $i,Y: $i] :
( ~ ( member @ X @ Z )
| ~ ( member @ Y @ Z )
| ( subclass @ ( unordered_pair @ X @ Y ) @ Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_is_subset) ).
thf(16,axiom,
! [X: $i,Y: $i,U: $i,V: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( ( unordered_pair @ X @ Y )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom3) ).
thf(17,axiom,
! [Y: $i,X: $i] :
( ~ ( member @ Y @ universal_class )
| ( ( unordered_pair @ X @ Y )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom2) ).
thf(18,axiom,
! [X: $i,Y: $i] :
( ~ ( member @ X @ universal_class )
| ( ( unordered_pair @ X @ Y )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom1) ).
thf(19,axiom,
! [X: $i,Z: $i,Y: $i] :
( ( ( unordered_pair @ X @ Z )
!= ( unordered_pair @ Y @ Z ) )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ( X = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',right_cancellation) ).
thf(20,axiom,
! [X: $i,Y: $i,Z: $i] :
( ( ( unordered_pair @ X @ Y )
!= ( unordered_pair @ X @ Z ) )
| ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( cross_product @ universal_class @ universal_class ) )
| ( Y = Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',left_cancellation) ).
thf(21,axiom,
! [X: $i,Y: $i] :
( ( ( unordered_pair @ X @ Y )
= null_class )
| ( member @ X @ universal_class )
| ( member @ Y @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_unordered_pair) ).
thf(22,axiom,
! [X: $i,Y: $i] :
( ( member @ X @ universal_class )
| ( ( unordered_pair @ X @ Y )
= ( singleton @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_equals_singleton2) ).
thf(23,axiom,
! [Y: $i,X: $i] :
( ( member @ Y @ universal_class )
| ( ( unordered_pair @ X @ Y )
= ( singleton @ X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_equals_singleton1) ).
thf(24,axiom,
! [Y: $i,X: $i] : ( subclass @ ( singleton @ Y ) @ ( unordered_pair @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair2) ).
thf(25,axiom,
! [X: $i,Y: $i] : ( subclass @ ( singleton @ X ) @ ( unordered_pair @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair1) ).
thf(26,axiom,
! [X: $i,Y: $i] :
( ( unordered_pair @ X @ Y )
= ( unordered_pair @ Y @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
thf(27,axiom,
member @ null_class @ universal_class,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_a_set) ).
thf(28,axiom,
! [Z: $i] :
( ( Z = null_class )
| ( member @ ( not_subclass_element @ Z @ null_class ) @ Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_unique) ).
thf(29,axiom,
! [X: $i] :
( ~ ( subclass @ X @ null_class )
| ( X = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_of_null_class_is_subclass) ).
thf(30,axiom,
! [X: $i] : ( subclass @ null_class @ X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_subclass) ).
thf(31,axiom,
! [Z: $i] :
~ ( member @ Z @ null_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_null_class) ).
thf(32,axiom,
! [Y: $i,X: $i] :
~ ( member @ Y @ ( intersection @ ( complement @ X ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',special_classes_lemma) ).
thf(33,axiom,
! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ~ ( member @ ( not_subclass_element @ Y @ X ) @ X )
| ( X = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality4) ).
thf(34,axiom,
! [Y: $i,X: $i] :
( ~ ( member @ ( not_subclass_element @ Y @ X ) @ X )
| ( X = Y )
| ( member @ ( not_subclass_element @ X @ Y ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality3) ).
thf(35,axiom,
! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ( X = Y )
| ( member @ ( not_subclass_element @ Y @ X ) @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality2) ).
thf(36,axiom,
! [X: $i,Y: $i] :
( ( X = Y )
| ( member @ ( not_subclass_element @ X @ Y ) @ X )
| ( member @ ( not_subclass_element @ Y @ X ) @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality1) ).
thf(37,axiom,
! [X: $i,Y: $i,Z: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( subclass @ Y @ Z )
| ( subclass @ X @ Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',transitivity_of_subclass) ).
thf(38,axiom,
! [X: $i] : ( subclass @ X @ X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_is_reflexive) ).
thf(39,axiom,
! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ V @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
thf(40,axiom,
! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ U @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_cartesian_product) ).
thf(41,axiom,
! [X: $i,Y: $i,U: $i,V: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( member @ Y @ ( unordered_pair @ X @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_unordered_pair) ).
thf(42,axiom,
! [X: $i,Y: $i,U: $i,V: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( member @ X @ ( unordered_pair @ X @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_unordered_pair) ).
thf(43,axiom,
! [Xf1: $i,Xf2: $i,Xh: $i] :
( ~ ( operation @ Xf1 )
| ~ ( operation @ Xf2 )
| ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( apply @ Xf2 @ ( ordered_pair @ ( apply @ Xh @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) ) @ ( apply @ Xh @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) ) )
!= ( apply @ Xh @ ( apply @ Xf1 @ ( ordered_pair @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) ) ) )
| ( homomorphism @ Xh @ Xf1 @ Xf2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism6) ).
thf(44,axiom,
! [Xf1: $i,Xf2: $i,Xh: $i] :
( ~ ( operation @ Xf1 )
| ~ ( operation @ Xf2 )
| ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) @ ( domain_of @ Xf1 ) )
| ( homomorphism @ Xh @ Xf1 @ Xf2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism5) ).
thf(45,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i,X: $i,Y: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( domain_of @ Xf1 ) )
| ( ( apply @ Xf2 @ ( ordered_pair @ ( apply @ Xh @ X ) @ ( apply @ Xh @ Y ) ) )
= ( apply @ Xh @ ( apply @ Xf1 @ ( ordered_pair @ X @ Y ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism4) ).
thf(46,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(47,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(48,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf1 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(49,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( function @ Xh )
| ( ( domain_of @ ( domain_of @ Xf1 ) )
!= ( domain_of @ Xh ) )
| ~ ( subclass @ ( range_of @ Xh ) @ ( domain_of @ ( domain_of @ Xf2 ) ) )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible4) ).
thf(50,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( subclass @ ( range_of @ Xh ) @ ( domain_of @ ( domain_of @ Xf2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible3) ).
thf(51,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( domain_of @ ( domain_of @ Xf1 ) )
= ( domain_of @ Xh ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible2) ).
thf(52,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( function @ Xh ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(53,axiom,
! [Xf: $i] :
( ~ ( function @ Xf )
| ( ( cross_product @ ( domain_of @ ( domain_of @ Xf ) ) @ ( domain_of @ ( domain_of @ Xf ) ) )
!= ( domain_of @ Xf ) )
| ~ ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) )
| ( operation @ Xf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation4) ).
thf(54,axiom,
! [Xf: $i] :
( ~ ( operation @ Xf )
| ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(55,axiom,
! [Xf: $i] :
( ~ ( operation @ Xf )
| ( ( cross_product @ ( domain_of @ ( domain_of @ Xf ) ) @ ( domain_of @ ( domain_of @ Xf ) ) )
= ( domain_of @ Xf ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation2) ).
thf(56,axiom,
! [Xf: $i] :
( ~ ( operation @ Xf )
| ( function @ Xf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(57,axiom,
! [X: $i] :
( ( intersection @ ( domain_of @ X ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ X ) ) )
= ( cantor @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cantor_class) ).
thf(58,axiom,
! [Xr: $i] :
( ( complement @ ( domain_of @ ( intersection @ Xr @ identity_relation ) ) )
= ( diagonalise @ Xr ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(59,axiom,
( ( intersection @ ( inverse @ subset_relation ) @ subset_relation )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(60,axiom,
( ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( complement @ ( compose @ ( complement @ element_relation ) @ ( inverse @ element_relation ) ) ) ) )
= subset_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset_relation) ).
thf(61,axiom,
! [Xf: $i] :
( ~ ( function @ ( inverse @ Xf ) )
| ~ ( function @ Xf )
| ( one_to_one @ Xf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one3) ).
thf(62,axiom,
! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ ( inverse @ Xf ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one2) ).
thf(63,axiom,
! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ Xf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one1) ).
thf(64,axiom,
! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( Y = null_class )
| ( member @ ( apply @ choice @ Y ) @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(65,axiom,
function @ choice,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(66,axiom,
! [Xf: $i,Y: $i] :
( ( sum_class @ ( image @ Xf @ ( singleton @ Y ) ) )
= ( apply @ Xf @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(67,axiom,
! [X: $i] :
( ( X = null_class )
| ( ( intersection @ X @ ( regular @ X ) )
= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(68,axiom,
! [X: $i] :
( ( X = null_class )
| ( member @ ( regular @ X ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(69,axiom,
! [Xf: $i,X: $i] :
( ~ ( function @ Xf )
| ~ ( member @ X @ universal_class )
| ( member @ ( image @ Xf @ X ) @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',replacement) ).
thf(70,axiom,
! [Xf: $i] :
( ~ ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation )
| ( function @ Xf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function3) ).
thf(71,axiom,
! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(72,axiom,
! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(73,axiom,
! [X: $i] :
( ~ ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation )
| ( single_valued_class @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class2) ).
thf(74,axiom,
! [X: $i] :
( ~ ( single_valued_class @ X )
| ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class1) ).
thf(75,axiom,
! [Z: $i,Yr: $i,Xr: $i,Y: $i] :
( ~ ( member @ Z @ ( image @ Yr @ ( image @ Xr @ ( singleton @ Y ) ) ) )
| ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ Y @ Z ) @ ( compose @ Yr @ Xr ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose3) ).
thf(76,axiom,
! [Y: $i,Z: $i,Yr: $i,Xr: $i] :
( ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( compose @ Yr @ Xr ) )
| ( member @ Z @ ( image @ Yr @ ( image @ Xr @ ( singleton @ Y ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose2) ).
thf(77,axiom,
! [Yr: $i,Xr: $i] : ( subclass @ ( compose @ Yr @ Xr ) @ ( cross_product @ universal_class @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(78,axiom,
! [U: $i] :
( ~ ( member @ U @ universal_class )
| ( member @ ( power_class @ U ) @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(79,axiom,
! [X: $i] :
( ( complement @ ( image @ element_relation @ ( complement @ X ) ) )
= ( power_class @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(80,axiom,
! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ ( sum_class @ X ) @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(81,axiom,
! [X: $i] :
( ( domain_of @ ( restrict @ element_relation @ universal_class @ X ) )
= ( sum_class @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class_definition) ).
thf(82,axiom,
member @ omega @ universal_class,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(83,axiom,
! [Y: $i] :
( ~ ( inductive @ Y )
| ( subclass @ omega @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(84,axiom,
inductive @ omega,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(85,axiom,
! [X: $i] :
( ~ ( member @ null_class @ X )
| ~ ( subclass @ ( image @ successor_relation @ X ) @ X )
| ( inductive @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(86,axiom,
! [X: $i] :
( ~ ( inductive @ X )
| ( subclass @ ( image @ successor_relation @ X ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(87,axiom,
! [X: $i] :
( ~ ( inductive @ X )
| ( member @ null_class @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(88,axiom,
! [X: $i,Y: $i] :
( ( ( successor @ X )
!= Y )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ X @ Y ) @ successor_relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation3) ).
thf(89,axiom,
! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ successor_relation )
| ( ( successor @ X )
= Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation2) ).
thf(90,axiom,
subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(91,axiom,
! [X: $i] :
( ( union @ X @ ( singleton @ X ) )
= ( successor @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(92,axiom,
! [Xr: $i,X: $i] :
( ( range_of @ ( restrict @ Xr @ X @ universal_class ) )
= ( image @ Xr @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(93,axiom,
! [Z: $i,X: $i,Y: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ Z @ ( singleton @ X ) @ Y ) @ null_class ) )
= ( range @ Z @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range) ).
thf(94,axiom,
! [Z: $i,X: $i,Y: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ Z @ X @ ( singleton @ Y ) ) @ null_class ) )
= ( domain @ Z @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain) ).
thf(95,axiom,
! [Z: $i] :
( ( domain_of @ ( inverse @ Z ) )
= ( range_of @ Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(96,axiom,
! [Y: $i] :
( ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) )
= ( inverse @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(97,axiom,
! [V: $i,U: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip3) ).
thf(98,axiom,
! [U: $i,V: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip2) ).
thf(99,axiom,
! [X: $i] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip1) ).
thf(100,axiom,
! [V: $i,W: $i,U: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate3) ).
thf(101,axiom,
! [U: $i,V: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate2) ).
thf(102,axiom,
! [X: $i] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate1) ).
thf(103,axiom,
! [Z: $i,X: $i] :
( ~ ( member @ Z @ universal_class )
| ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
= null_class )
| ( member @ Z @ ( domain_of @ X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain2) ).
thf(104,axiom,
! [X: $i,Z: $i] :
( ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
!= null_class )
| ~ ( member @ Z @ ( domain_of @ X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain1) ).
thf(105,axiom,
! [X: $i,Y: $i,Xr: $i] :
( ( intersection @ ( cross_product @ X @ Y ) @ Xr )
= ( restrict @ Xr @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction2) ).
thf(106,axiom,
! [Xr: $i,X: $i,Y: $i] :
( ( intersection @ Xr @ ( cross_product @ X @ Y ) )
= ( restrict @ Xr @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction1) ).
thf(107,axiom,
! [X: $i,Y: $i] :
( ( intersection @ ( complement @ ( intersection @ X @ Y ) ) @ ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) ) )
= ( symmetric_difference @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetric_difference) ).
thf(108,axiom,
! [X: $i,Y: $i] :
( ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) )
= ( union @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(109,axiom,
! [Z: $i,X: $i] :
( ~ ( member @ Z @ universal_class )
| ( member @ Z @ ( complement @ X ) )
| ( member @ Z @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement2) ).
thf(110,axiom,
! [Z: $i,X: $i] :
( ~ ( member @ Z @ ( complement @ X ) )
| ~ ( member @ Z @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(111,axiom,
! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ X )
| ~ ( member @ Z @ Y )
| ( member @ Z @ ( intersection @ X @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection3) ).
thf(112,axiom,
! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(113,axiom,
! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(114,axiom,
! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ X @ Y )
| ( member @ ( ordered_pair @ X @ Y ) @ element_relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation3) ).
thf(115,axiom,
! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ element_relation )
| ( member @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation2) ).
thf(116,axiom,
subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(117,axiom,
! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( cross_product @ X @ Y ) )
| ( ( ordered_pair @ ( first @ Z ) @ ( second @ Z ) )
= Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product4) ).
thf(118,axiom,
! [U: $i,X: $i,V: $i,Y: $i] :
( ~ ( member @ U @ X )
| ~ ( member @ V @ Y )
| ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product3) ).
thf(119,axiom,
! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ V @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product2) ).
thf(120,axiom,
! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ U @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product1) ).
thf(121,axiom,
! [X: $i,Y: $i] :
( ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) )
= ( ordered_pair @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair) ).
thf(122,axiom,
! [X: $i] :
( ( unordered_pair @ X @ X )
= ( singleton @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(123,axiom,
! [X: $i,Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(124,axiom,
! [Y: $i,X: $i] :
( ~ ( member @ Y @ universal_class )
| ( member @ Y @ ( unordered_pair @ X @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair3) ).
thf(125,axiom,
! [X: $i,Y: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ X @ ( unordered_pair @ X @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair2) ).
thf(126,axiom,
! [U: $i,X: $i,Y: $i] :
( ~ ( member @ U @ ( unordered_pair @ X @ Y ) )
| ( U = X )
| ( U = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(127,axiom,
! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( subclass @ Y @ X )
| ( X = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(128,axiom,
! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ Y @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(129,axiom,
! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(130,axiom,
! [X: $i] : ( subclass @ X @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(131,axiom,
! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ( subclass @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(132,axiom,
! [X: $i,Y: $i] :
( ( member @ ( not_subclass_element @ X @ Y ) @ X )
| ( subclass @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(133,axiom,
! [X: $i,Y: $i,U: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( member @ U @ X )
| ( member @ U @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_members) ).
thf(134,conjecture,
$false,
file('no conjecture given, we try to refute the axioms',dummy_conjecture) ).
thf(135,negated_conjecture,
$false = $false,
inference(negate_conjecture,[status(cth)],[134]) ).
thf(136,negated_conjecture,
( ( member_of @ ( singleton @ u ) )
!= u ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_member_of_singleton_is_unique_2) ).
thf(137,negated_conjecture,
member @ u @ universal_class,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_member_of_singleton_is_unique_1) ).
thf(138,plain,
$false = $false,
inference(unfold_def,[status(thm)],[135]) ).
thf(139,plain,
( ( ! [X: $i] :
( ( ( singleton @ ( member_of @ X ) )
= X )
| ( ( member_of @ X )
= X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[1]) ).
thf(140,plain,
( ( ! [X: $i] :
( ( member @ ( member_of @ X ) @ universal_class )
| ( ( member_of @ X )
= X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[2]) ).
thf(141,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( ( singleton @ ( member_of @ ( singleton @ Y ) ) )
= ( singleton @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[3]) ).
thf(142,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( member @ ( member_of @ ( singleton @ Y ) ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[4]) ).
thf(143,plain,
( ( ! [Y: $i,Z: $i,X: $i] :
( ( ( unordered_pair @ Y @ Z )
!= ( singleton @ X ) )
| ~ ( member @ X @ universal_class )
| ( X = Y )
| ( X = Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[5]) ).
thf(144,plain,
( ( ! [X: $i,Y: $i] :
( ( ( singleton @ X )
!= ( singleton @ Y ) )
| ~ ( member @ Y @ universal_class )
| ( X = Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[6]) ).
thf(145,plain,
( ( ! [X: $i,Y: $i] :
( ( ( singleton @ X )
!= ( singleton @ Y ) )
| ~ ( member @ X @ universal_class )
| ( X = Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[7]) ).
thf(146,plain,
( ( ! [X: $i] :
( ( member @ X @ universal_class )
| ( ( singleton @ X )
= null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[8]) ).
thf(147,plain,
( ( ! [Y: $i,X: $i] :
( ~ ( member @ Y @ ( singleton @ X ) )
| ( Y = X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[9]) ).
thf(148,plain,
( ( member @ null_class @ ( singleton @ null_class ) )
= $true ),
inference(unfold_def,[status(thm)],[10]) ).
thf(149,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( ( singleton @ X )
!= null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[11]) ).
thf(150,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ X @ ( singleton @ X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[12]) ).
thf(151,plain,
( ( ! [Y: $i,X: $i] : ( member @ ( singleton @ Y ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[13]) ).
thf(152,plain,
( ( ! [X: $i] : ( member @ ( singleton @ X ) @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[14]) ).
thf(153,plain,
( ( ! [X: $i,Z: $i,Y: $i] :
( ~ ( member @ X @ Z )
| ~ ( member @ Y @ Z )
| ( subclass @ ( unordered_pair @ X @ Y ) @ Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[15]) ).
thf(154,plain,
( ( ! [X: $i,Y: $i,U: $i,V: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[16]) ).
thf(155,plain,
( ( ! [Y: $i,X: $i] :
( ~ ( member @ Y @ universal_class )
| ( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[17]) ).
thf(156,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ X @ universal_class )
| ( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[18]) ).
thf(157,plain,
( ( ! [X: $i,Z: $i,Y: $i] :
( ( ( unordered_pair @ X @ Z )
!= ( unordered_pair @ Y @ Z ) )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ( X = Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[19]) ).
thf(158,plain,
( ( ! [X: $i,Y: $i,Z: $i] :
( ( ( unordered_pair @ X @ Y )
!= ( unordered_pair @ X @ Z ) )
| ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( cross_product @ universal_class @ universal_class ) )
| ( Y = Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[20]) ).
thf(159,plain,
( ( ! [X: $i,Y: $i] :
( ( ( unordered_pair @ X @ Y )
= null_class )
| ( member @ X @ universal_class )
| ( member @ Y @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[21]) ).
thf(160,plain,
( ( ! [X: $i,Y: $i] :
( ( member @ X @ universal_class )
| ( ( unordered_pair @ X @ Y )
= ( singleton @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[22]) ).
thf(161,plain,
( ( ! [Y: $i,X: $i] :
( ( member @ Y @ universal_class )
| ( ( unordered_pair @ X @ Y )
= ( singleton @ X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[23]) ).
thf(162,plain,
( ( ! [Y: $i,X: $i] : ( subclass @ ( singleton @ Y ) @ ( unordered_pair @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[24]) ).
thf(163,plain,
( ( ! [X: $i,Y: $i] : ( subclass @ ( singleton @ X ) @ ( unordered_pair @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[25]) ).
thf(164,plain,
( ( ! [X: $i,Y: $i] :
( ( unordered_pair @ X @ Y )
= ( unordered_pair @ Y @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[26]) ).
thf(165,plain,
( ( member @ null_class @ universal_class )
= $true ),
inference(unfold_def,[status(thm)],[27]) ).
thf(166,plain,
( ( ! [Z: $i] :
( ( Z = null_class )
| ( member @ ( not_subclass_element @ Z @ null_class ) @ Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[28]) ).
thf(167,plain,
( ( ! [X: $i] :
( ~ ( subclass @ X @ null_class )
| ( X = null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[29]) ).
thf(168,plain,
( ( ! [X: $i] : ( subclass @ null_class @ X ) )
= $true ),
inference(unfold_def,[status(thm)],[30]) ).
thf(169,plain,
( ( ! [Z: $i] :
~ ( member @ Z @ null_class ) )
= $true ),
inference(unfold_def,[status(thm)],[31]) ).
thf(170,plain,
( ( ! [Y: $i,X: $i] :
~ ( member @ Y @ ( intersection @ ( complement @ X ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[32]) ).
thf(171,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ~ ( member @ ( not_subclass_element @ Y @ X ) @ X )
| ( X = Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[33]) ).
thf(172,plain,
( ( ! [Y: $i,X: $i] :
( ~ ( member @ ( not_subclass_element @ Y @ X ) @ X )
| ( X = Y )
| ( member @ ( not_subclass_element @ X @ Y ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[34]) ).
thf(173,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ( X = Y )
| ( member @ ( not_subclass_element @ Y @ X ) @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[35]) ).
thf(174,plain,
( ( ! [X: $i,Y: $i] :
( ( X = Y )
| ( member @ ( not_subclass_element @ X @ Y ) @ X )
| ( member @ ( not_subclass_element @ Y @ X ) @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[36]) ).
thf(175,plain,
( ( ! [X: $i,Y: $i,Z: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( subclass @ Y @ Z )
| ( subclass @ X @ Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[37]) ).
thf(176,plain,
( ( ! [X: $i] : ( subclass @ X @ X ) )
= $true ),
inference(unfold_def,[status(thm)],[38]) ).
thf(177,plain,
( ( ! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ V @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[39]) ).
thf(178,plain,
( ( ! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ U @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[40]) ).
thf(179,plain,
( ( ! [X: $i,Y: $i,U: $i,V: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( member @ Y @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[41]) ).
thf(180,plain,
( ( ! [X: $i,Y: $i,U: $i,V: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( member @ X @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[42]) ).
thf(181,plain,
( ( ! [Xf1: $i,Xf2: $i,Xh: $i] :
( ~ ( operation @ Xf1 )
| ~ ( operation @ Xf2 )
| ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( apply @ Xf2 @ ( ordered_pair @ ( apply @ Xh @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) ) @ ( apply @ Xh @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) ) )
!= ( apply @ Xh @ ( apply @ Xf1 @ ( ordered_pair @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) ) ) )
| ( homomorphism @ Xh @ Xf1 @ Xf2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[43]) ).
thf(182,plain,
( ( ! [Xf1: $i,Xf2: $i,Xh: $i] :
( ~ ( operation @ Xf1 )
| ~ ( operation @ Xf2 )
| ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) @ ( domain_of @ Xf1 ) )
| ( homomorphism @ Xh @ Xf1 @ Xf2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[44]) ).
thf(183,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i,X: $i,Y: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( domain_of @ Xf1 ) )
| ( ( apply @ Xf2 @ ( ordered_pair @ ( apply @ Xh @ X ) @ ( apply @ Xh @ Y ) ) )
= ( apply @ Xh @ ( apply @ Xf1 @ ( ordered_pair @ X @ Y ) ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[45]) ).
thf(184,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[46]) ).
thf(185,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[47]) ).
thf(186,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[48]) ).
thf(187,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( function @ Xh )
| ( ( domain_of @ ( domain_of @ Xf1 ) )
!= ( domain_of @ Xh ) )
| ~ ( subclass @ ( range_of @ Xh ) @ ( domain_of @ ( domain_of @ Xf2 ) ) )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[49]) ).
thf(188,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( subclass @ ( range_of @ Xh ) @ ( domain_of @ ( domain_of @ Xf2 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[50]) ).
thf(189,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( domain_of @ ( domain_of @ Xf1 ) )
= ( domain_of @ Xh ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[51]) ).
thf(190,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( function @ Xh ) ) )
= $true ),
inference(unfold_def,[status(thm)],[52]) ).
thf(191,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( ( cross_product @ ( domain_of @ ( domain_of @ Xf ) ) @ ( domain_of @ ( domain_of @ Xf ) ) )
!= ( domain_of @ Xf ) )
| ~ ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) )
| ( operation @ Xf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[53]) ).
thf(192,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[54]) ).
thf(193,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( ( cross_product @ ( domain_of @ ( domain_of @ Xf ) ) @ ( domain_of @ ( domain_of @ Xf ) ) )
= ( domain_of @ Xf ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[55]) ).
thf(194,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( function @ Xf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[56]) ).
thf(195,plain,
( ( ! [X: $i] :
( ( intersection @ ( domain_of @ X ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ X ) ) )
= ( cantor @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[57]) ).
thf(196,plain,
( ( ! [Xr: $i] :
( ( complement @ ( domain_of @ ( intersection @ Xr @ identity_relation ) ) )
= ( diagonalise @ Xr ) ) )
= $true ),
inference(unfold_def,[status(thm)],[58]) ).
thf(197,plain,
( ( ( intersection @ ( inverse @ subset_relation ) @ subset_relation )
= identity_relation )
= $true ),
inference(unfold_def,[status(thm)],[59]) ).
thf(198,plain,
( ( ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( complement @ ( compose @ ( complement @ element_relation ) @ ( inverse @ element_relation ) ) ) ) )
= subset_relation )
= $true ),
inference(unfold_def,[status(thm)],[60]) ).
thf(199,plain,
( ( ! [Xf: $i] :
( ~ ( function @ ( inverse @ Xf ) )
| ~ ( function @ Xf )
| ( one_to_one @ Xf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[61]) ).
thf(200,plain,
( ( ! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ ( inverse @ Xf ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[62]) ).
thf(201,plain,
( ( ! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ Xf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[63]) ).
thf(202,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( Y = null_class )
| ( member @ ( apply @ choice @ Y ) @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[64]) ).
thf(203,plain,
( ( function @ choice )
= $true ),
inference(unfold_def,[status(thm)],[65]) ).
thf(204,plain,
( ( ! [Xf: $i,Y: $i] :
( ( sum_class @ ( image @ Xf @ ( singleton @ Y ) ) )
= ( apply @ Xf @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[66]) ).
thf(205,plain,
( ( ! [X: $i] :
( ( X = null_class )
| ( ( intersection @ X @ ( regular @ X ) )
= null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[67]) ).
thf(206,plain,
( ( ! [X: $i] :
( ( X = null_class )
| ( member @ ( regular @ X ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[68]) ).
thf(207,plain,
( ( ! [Xf: $i,X: $i] :
( ~ ( function @ Xf )
| ~ ( member @ X @ universal_class )
| ( member @ ( image @ Xf @ X ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[69]) ).
thf(208,plain,
( ( ! [Xf: $i] :
( ~ ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation )
| ( function @ Xf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[70]) ).
thf(209,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation ) ) )
= $true ),
inference(unfold_def,[status(thm)],[71]) ).
thf(210,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[72]) ).
thf(211,plain,
( ( ! [X: $i] :
( ~ ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation )
| ( single_valued_class @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[73]) ).
thf(212,plain,
( ( ! [X: $i] :
( ~ ( single_valued_class @ X )
| ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation ) ) )
= $true ),
inference(unfold_def,[status(thm)],[74]) ).
thf(213,plain,
( ( ! [Z: $i,Yr: $i,Xr: $i,Y: $i] :
( ~ ( member @ Z @ ( image @ Yr @ ( image @ Xr @ ( singleton @ Y ) ) ) )
| ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ Y @ Z ) @ ( compose @ Yr @ Xr ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[75]) ).
thf(214,plain,
( ( ! [Y: $i,Z: $i,Yr: $i,Xr: $i] :
( ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( compose @ Yr @ Xr ) )
| ( member @ Z @ ( image @ Yr @ ( image @ Xr @ ( singleton @ Y ) ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[76]) ).
thf(215,plain,
( ( ! [Yr: $i,Xr: $i] : ( subclass @ ( compose @ Yr @ Xr ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[77]) ).
thf(216,plain,
( ( ! [U: $i] :
( ~ ( member @ U @ universal_class )
| ( member @ ( power_class @ U ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[78]) ).
thf(217,plain,
( ( ! [X: $i] :
( ( complement @ ( image @ element_relation @ ( complement @ X ) ) )
= ( power_class @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[79]) ).
thf(218,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ ( sum_class @ X ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[80]) ).
thf(219,plain,
( ( ! [X: $i] :
( ( domain_of @ ( restrict @ element_relation @ universal_class @ X ) )
= ( sum_class @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[81]) ).
thf(220,plain,
( ( member @ omega @ universal_class )
= $true ),
inference(unfold_def,[status(thm)],[82]) ).
thf(221,plain,
( ( ! [Y: $i] :
( ~ ( inductive @ Y )
| ( subclass @ omega @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[83]) ).
thf(222,plain,
( ( inductive @ omega )
= $true ),
inference(unfold_def,[status(thm)],[84]) ).
thf(223,plain,
( ( ! [X: $i] :
( ~ ( member @ null_class @ X )
| ~ ( subclass @ ( image @ successor_relation @ X ) @ X )
| ( inductive @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[85]) ).
thf(224,plain,
( ( ! [X: $i] :
( ~ ( inductive @ X )
| ( subclass @ ( image @ successor_relation @ X ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[86]) ).
thf(225,plain,
( ( ! [X: $i] :
( ~ ( inductive @ X )
| ( member @ null_class @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[87]) ).
thf(226,plain,
( ( ! [X: $i,Y: $i] :
( ( ( successor @ X )
!= Y )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ X @ Y ) @ successor_relation ) ) )
= $true ),
inference(unfold_def,[status(thm)],[88]) ).
thf(227,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ successor_relation )
| ( ( successor @ X )
= Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[89]) ).
thf(228,plain,
( ( subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[90]) ).
thf(229,plain,
( ( ! [X: $i] :
( ( union @ X @ ( singleton @ X ) )
= ( successor @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[91]) ).
thf(230,plain,
( ( ! [Xr: $i,X: $i] :
( ( range_of @ ( restrict @ Xr @ X @ universal_class ) )
= ( image @ Xr @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[92]) ).
thf(231,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ Z @ ( singleton @ X ) @ Y ) @ null_class ) )
= ( range @ Z @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[93]) ).
thf(232,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ Z @ X @ ( singleton @ Y ) ) @ null_class ) )
= ( domain @ Z @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[94]) ).
thf(233,plain,
( ( ! [Z: $i] :
( ( domain_of @ ( inverse @ Z ) )
= ( range_of @ Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[95]) ).
thf(234,plain,
( ( ! [Y: $i] :
( ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) )
= ( inverse @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[96]) ).
thf(235,plain,
( ( ! [V: $i,U: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[97]) ).
thf(236,plain,
( ( ! [U: $i,V: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[98]) ).
thf(237,plain,
( ( ! [X: $i] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[99]) ).
thf(238,plain,
( ( ! [V: $i,W: $i,U: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[100]) ).
thf(239,plain,
( ( ! [U: $i,V: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[101]) ).
thf(240,plain,
( ( ! [X: $i] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[102]) ).
thf(241,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ universal_class )
| ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
= null_class )
| ( member @ Z @ ( domain_of @ X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[103]) ).
thf(242,plain,
( ( ! [X: $i,Z: $i] :
( ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
!= null_class )
| ~ ( member @ Z @ ( domain_of @ X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[104]) ).
thf(243,plain,
( ( ! [X: $i,Y: $i,Xr: $i] :
( ( intersection @ ( cross_product @ X @ Y ) @ Xr )
= ( restrict @ Xr @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[105]) ).
thf(244,plain,
( ( ! [Xr: $i,X: $i,Y: $i] :
( ( intersection @ Xr @ ( cross_product @ X @ Y ) )
= ( restrict @ Xr @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[106]) ).
thf(245,plain,
( ( ! [X: $i,Y: $i] :
( ( intersection @ ( complement @ ( intersection @ X @ Y ) ) @ ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) ) )
= ( symmetric_difference @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[107]) ).
thf(246,plain,
( ( ! [X: $i,Y: $i] :
( ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) )
= ( union @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[108]) ).
thf(247,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ universal_class )
| ( member @ Z @ ( complement @ X ) )
| ( member @ Z @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[109]) ).
thf(248,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ ( complement @ X ) )
| ~ ( member @ Z @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[110]) ).
thf(249,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ X )
| ~ ( member @ Z @ Y )
| ( member @ Z @ ( intersection @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[111]) ).
thf(250,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[112]) ).
thf(251,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[113]) ).
thf(252,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ X @ Y )
| ( member @ ( ordered_pair @ X @ Y ) @ element_relation ) ) )
= $true ),
inference(unfold_def,[status(thm)],[114]) ).
thf(253,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ element_relation )
| ( member @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[115]) ).
thf(254,plain,
( ( subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[116]) ).
thf(255,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( cross_product @ X @ Y ) )
| ( ( ordered_pair @ ( first @ Z ) @ ( second @ Z ) )
= Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[117]) ).
thf(256,plain,
( ( ! [U: $i,X: $i,V: $i,Y: $i] :
( ~ ( member @ U @ X )
| ~ ( member @ V @ Y )
| ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[118]) ).
thf(257,plain,
( ( ! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ V @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[119]) ).
thf(258,plain,
( ( ! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ U @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[120]) ).
thf(259,plain,
( ( ! [X: $i,Y: $i] :
( ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) )
= ( ordered_pair @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[121]) ).
thf(260,plain,
( ( ! [X: $i] :
( ( unordered_pair @ X @ X )
= ( singleton @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[122]) ).
thf(261,plain,
( ( ! [X: $i,Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[123]) ).
thf(262,plain,
( ( ! [Y: $i,X: $i] :
( ~ ( member @ Y @ universal_class )
| ( member @ Y @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[124]) ).
thf(263,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ X @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[125]) ).
thf(264,plain,
( ( ! [U: $i,X: $i,Y: $i] :
( ~ ( member @ U @ ( unordered_pair @ X @ Y ) )
| ( U = X )
| ( U = Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[126]) ).
thf(265,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( subclass @ Y @ X )
| ( X = Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[127]) ).
thf(266,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ Y @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[128]) ).
thf(267,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[129]) ).
thf(268,plain,
( ( ! [X: $i] : ( subclass @ X @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[130]) ).
thf(269,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[131]) ).
thf(270,plain,
( ( ! [X: $i,Y: $i] :
( ( member @ ( not_subclass_element @ X @ Y ) @ X )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[132]) ).
thf(271,plain,
( ( ! [X: $i,Y: $i,U: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( member @ U @ X )
| ( member @ U @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[133]) ).
thf(272,plain,
( ( ( member_of @ ( singleton @ u ) )
!= u )
= $true ),
inference(unfold_def,[status(thm)],[136]) ).
thf(273,plain,
( ( member @ u @ universal_class )
= $true ),
inference(unfold_def,[status(thm)],[137]) ).
thf(274,plain,
( ( ~ $false ) = $true ),
inference(polarity_switch,[status(thm)],[138]) ).
thf(275,plain,
( ( ! [X: $i] :
( ( ( member_of @ X )
= X )
| ( member @ ( member_of @ X ) @ universal_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[140]) ).
thf(276,plain,
( ( ! [Y: $i,Z: $i,X: $i] :
( ( ( unordered_pair @ Y @ Z )
!= ( singleton @ X ) )
| ~ ( member @ X @ universal_class )
| ( X = Y )
| ( X = Z ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[143]) ).
thf(277,plain,
( ( ! [X: $i,Y: $i] :
( ( ( singleton @ X )
!= ( singleton @ Y ) )
| ~ ( member @ Y @ universal_class )
| ( X = Y ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[144]) ).
thf(278,plain,
( ( ! [X: $i,Y: $i] :
( ( ( singleton @ X )
!= ( singleton @ Y ) )
| ~ ( member @ X @ universal_class )
| ( X = Y ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[145]) ).
thf(279,plain,
( ( ! [X: $i] :
( ( ( singleton @ X )
= null_class )
| ( member @ X @ universal_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[146]) ).
thf(280,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( ( singleton @ X )
!= null_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[149]) ).
thf(281,plain,
( ( ! [X: $i,Z: $i] :
( ~ ( member @ X @ Z )
| ! [Y: $i] :
( ~ ( member @ Y @ Z )
| ( subclass @ ( unordered_pair @ X @ Y ) @ Z ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[153]) ).
thf(282,plain,
( ( ! [X: $i,Y: $i] :
( ! [U: $i,V: $i] :
~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[154]) ).
thf(283,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ! [X: $i] :
( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[155]) ).
thf(284,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ! [Y: $i] :
( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[156]) ).
thf(285,plain,
( ( ! [X: $i,Z: $i,Y: $i] :
( ( ( unordered_pair @ X @ Z )
!= ( unordered_pair @ Y @ Z ) )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ( X = Y ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[157]) ).
thf(286,plain,
( ( ! [X: $i,Y: $i,Z: $i] :
( ( ( unordered_pair @ X @ Y )
!= ( unordered_pair @ X @ Z ) )
| ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( cross_product @ universal_class @ universal_class ) )
| ( Y = Z ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[158]) ).
thf(287,plain,
( ( ! [X: $i] :
( ! [Y: $i] :
( ( unordered_pair @ X @ Y )
= ( singleton @ Y ) )
| ( member @ X @ universal_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[160]) ).
thf(288,plain,
( ( ! [Y: $i] :
( ! [X: $i] :
( ( unordered_pair @ X @ Y )
= ( singleton @ X ) )
| ( member @ Y @ universal_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[161]) ).
thf(289,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ! [Z: $i] :
( ~ ( subclass @ Y @ Z )
| ( subclass @ X @ Z ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[175]) ).
thf(290,plain,
( ( ! [U: $i,V: $i] :
( ! [X: $i,Y: $i] :
~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ V @ universal_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[177]) ).
thf(291,plain,
( ( ! [U: $i,V: $i] :
( ! [X: $i,Y: $i] :
~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ U @ universal_class ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[178]) ).
thf(292,plain,
( ( ! [X: $i,Y: $i] :
( ! [U: $i,V: $i] :
~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( member @ Y @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[179]) ).
thf(293,plain,
( ( ! [X: $i,Y: $i] :
( ! [U: $i,V: $i] :
~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( member @ X @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[180]) ).
thf(294,plain,
( ( ! [Xf1: $i] :
( ~ ( operation @ Xf1 )
| ! [Xf2: $i] :
( ~ ( operation @ Xf2 )
| ! [Xh: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( apply @ Xf2 @ ( ordered_pair @ ( apply @ Xh @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) ) @ ( apply @ Xh @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) ) )
!= ( apply @ Xh @ ( apply @ Xf1 @ ( ordered_pair @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) ) ) )
| ( homomorphism @ Xh @ Xf1 @ Xf2 ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[181]) ).
thf(295,plain,
( ( ! [Xf1: $i] :
( ~ ( operation @ Xf1 )
| ! [Xf2: $i] :
( ~ ( operation @ Xf2 )
| ! [Xh: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) @ ( domain_of @ Xf1 ) ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[182]) ).
thf(296,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( domain_of @ Xf1 ) )
| ( ( apply @ Xf2 @ ( ordered_pair @ ( apply @ Xh @ X ) @ ( apply @ Xh @ Y ) ) )
= ( apply @ Xh @ ( apply @ Xf1 @ ( ordered_pair @ X @ Y ) ) ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[183]) ).
thf(297,plain,
( ( ! [Xh: $i,Xf1: $i] :
( ! [Xf2: $i] :
~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf1 ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[186]) ).
thf(298,plain,
( ( ! [Xh: $i] :
( ~ ( function @ Xh )
| ! [Xf1: $i] :
( ( ( domain_of @ ( domain_of @ Xf1 ) )
!= ( domain_of @ Xh ) )
| ! [Xf2: $i] :
( ~ ( subclass @ ( range_of @ Xh ) @ ( domain_of @ ( domain_of @ Xf2 ) ) )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[187]) ).
thf(299,plain,
( ( ! [Xh: $i,Xf1: $i] :
( ! [Xf2: $i] :
~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( domain_of @ ( domain_of @ Xf1 ) )
= ( domain_of @ Xh ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[189]) ).
thf(300,plain,
( ( ! [Xh: $i] :
( ! [Xf1: $i,Xf2: $i] :
~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( function @ Xh ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[190]) ).
thf(301,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( ( cross_product @ ( domain_of @ ( domain_of @ Xf ) ) @ ( domain_of @ ( domain_of @ Xf ) ) )
!= ( domain_of @ Xf ) )
| ~ ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) )
| ( operation @ Xf ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[191]) ).
thf(302,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ ( image @ Xf @ X ) @ universal_class ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[207]) ).
thf(303,plain,
( ( ! [X: $i,Y: $i] :
( ( ( successor @ X )
!= Y )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ X @ Y ) @ successor_relation ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[226]) ).
thf(304,plain,
( ( ! [Z: $i] :
( ~ ( member @ Z @ universal_class )
| ! [X: $i] :
( ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
= null_class )
| ( member @ Z @ ( domain_of @ X ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[241]) ).
thf(305,plain,
( ( ! [X: $i,Z: $i] :
( ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
!= null_class )
| ~ ( member @ Z @ ( domain_of @ X ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[242]) ).
thf(306,plain,
( ( ! [Z: $i] :
( ~ ( member @ Z @ universal_class )
| ! [X: $i] :
( ( member @ Z @ ( complement @ X ) )
| ( member @ Z @ X ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[247]) ).
thf(307,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ X )
| ! [Y: $i] :
( ~ ( member @ Z @ Y )
| ( member @ Z @ ( intersection @ X @ Y ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[249]) ).
thf(308,plain,
( ( ! [Z: $i,X: $i] :
( ! [Y: $i] :
~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[251]) ).
thf(309,plain,
( ( ! [Z: $i] :
( ! [X: $i,Y: $i] :
~ ( member @ Z @ ( cross_product @ X @ Y ) )
| ( ( ordered_pair @ ( first @ Z ) @ ( second @ Z ) )
= Z ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[255]) ).
thf(310,plain,
( ( ! [U: $i,X: $i] :
( ~ ( member @ U @ X )
| ! [V: $i,Y: $i] :
( ~ ( member @ V @ Y )
| ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[256]) ).
thf(311,plain,
( ( ! [U: $i,V: $i,X: $i] :
( ! [Y: $i] :
~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ U @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[258]) ).
thf(312,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ! [X: $i] : ( member @ Y @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[262]) ).
thf(313,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ! [Y: $i] : ( member @ X @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[263]) ).
thf(314,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ Y @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[266]) ).
thf(315,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[267]) ).
thf(316,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ! [U: $i] :
( ~ ( member @ U @ X )
| ( member @ U @ Y ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[271]) ).
thf(317,plain,
( ( ( member_of @ ( singleton @ u ) )
!= u )
= $true ),
inference(extcnf_combined,[status(esa)],[272]) ).
thf(318,plain,
( ( member @ u @ universal_class )
= $true ),
inference(copy,[status(thm)],[273]) ).
thf(319,plain,
( ( ( member_of @ ( singleton @ u ) )
!= u )
= $true ),
inference(copy,[status(thm)],[317]) ).
thf(320,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ! [U: $i] :
( ~ ( member @ U @ X )
| ( member @ U @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[316]) ).
thf(321,plain,
( ( ! [X: $i,Y: $i] :
( ( member @ ( not_subclass_element @ X @ Y ) @ X )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[270]) ).
thf(322,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[269]) ).
thf(323,plain,
( ( ! [X: $i] : ( subclass @ X @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[268]) ).
thf(324,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[315]) ).
thf(325,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ Y @ X ) ) )
= $true ),
inference(copy,[status(thm)],[314]) ).
thf(326,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( subclass @ Y @ X )
| ( X = Y ) ) )
= $true ),
inference(copy,[status(thm)],[265]) ).
thf(327,plain,
( ( ! [U: $i,X: $i,Y: $i] :
( ~ ( member @ U @ ( unordered_pair @ X @ Y ) )
| ( U = X )
| ( U = Y ) ) )
= $true ),
inference(copy,[status(thm)],[264]) ).
thf(328,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ! [Y: $i] : ( member @ X @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[313]) ).
thf(329,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ! [X: $i] : ( member @ Y @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[312]) ).
thf(330,plain,
( ( ! [X: $i,Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[261]) ).
thf(331,plain,
( ( ! [X: $i] :
( ( unordered_pair @ X @ X )
= ( singleton @ X ) ) )
= $true ),
inference(copy,[status(thm)],[260]) ).
thf(332,plain,
( ( ! [X: $i,Y: $i] :
( ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) )
= ( ordered_pair @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[259]) ).
thf(333,plain,
( ( ! [U: $i,V: $i,X: $i] :
( ! [Y: $i] :
~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ U @ X ) ) )
= $true ),
inference(copy,[status(thm)],[311]) ).
thf(334,plain,
( ( ! [U: $i,V: $i,X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ V @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[257]) ).
thf(335,plain,
( ( ! [U: $i,X: $i] :
( ~ ( member @ U @ X )
| ! [V: $i,Y: $i] :
( ~ ( member @ V @ Y )
| ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[310]) ).
thf(336,plain,
( ( ! [Z: $i] :
( ! [X: $i,Y: $i] :
~ ( member @ Z @ ( cross_product @ X @ Y ) )
| ( ( ordered_pair @ ( first @ Z ) @ ( second @ Z ) )
= Z ) ) )
= $true ),
inference(copy,[status(thm)],[309]) ).
thf(337,plain,
( ( subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[254]) ).
thf(338,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ element_relation )
| ( member @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[253]) ).
thf(339,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ X @ Y )
| ( member @ ( ordered_pair @ X @ Y ) @ element_relation ) ) )
= $true ),
inference(copy,[status(thm)],[252]) ).
thf(340,plain,
( ( ! [Z: $i,X: $i] :
( ! [Y: $i] :
~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ X ) ) )
= $true ),
inference(copy,[status(thm)],[308]) ).
thf(341,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[250]) ).
thf(342,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ X )
| ! [Y: $i] :
( ~ ( member @ Z @ Y )
| ( member @ Z @ ( intersection @ X @ Y ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[307]) ).
thf(343,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ ( complement @ X ) )
| ~ ( member @ Z @ X ) ) )
= $true ),
inference(copy,[status(thm)],[248]) ).
thf(344,plain,
( ( ! [Z: $i] :
( ~ ( member @ Z @ universal_class )
| ! [X: $i] :
( ( member @ Z @ ( complement @ X ) )
| ( member @ Z @ X ) ) ) )
= $true ),
inference(copy,[status(thm)],[306]) ).
thf(345,plain,
( ( ! [X: $i,Y: $i] :
( ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) )
= ( union @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[246]) ).
thf(346,plain,
( ( ! [X: $i,Y: $i] :
( ( intersection @ ( complement @ ( intersection @ X @ Y ) ) @ ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) ) )
= ( symmetric_difference @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[245]) ).
thf(347,plain,
( ( ! [Xr: $i,X: $i,Y: $i] :
( ( intersection @ Xr @ ( cross_product @ X @ Y ) )
= ( restrict @ Xr @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[244]) ).
thf(348,plain,
( ( ! [X: $i,Y: $i,Xr: $i] :
( ( intersection @ ( cross_product @ X @ Y ) @ Xr )
= ( restrict @ Xr @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[243]) ).
thf(349,plain,
( ( ! [X: $i,Z: $i] :
( ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
!= null_class )
| ~ ( member @ Z @ ( domain_of @ X ) ) ) )
= $true ),
inference(copy,[status(thm)],[305]) ).
thf(350,plain,
( ( ! [Z: $i] :
( ~ ( member @ Z @ universal_class )
| ! [X: $i] :
( ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
= null_class )
| ( member @ Z @ ( domain_of @ X ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[304]) ).
thf(351,plain,
( ( ! [X: $i] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[240]) ).
thf(352,plain,
( ( ! [U: $i,V: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[239]) ).
thf(353,plain,
( ( ! [V: $i,W: $i,U: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) ) ) )
= $true ),
inference(copy,[status(thm)],[238]) ).
thf(354,plain,
( ( ! [X: $i] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[237]) ).
thf(355,plain,
( ( ! [U: $i,V: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[236]) ).
thf(356,plain,
( ( ! [V: $i,U: $i,W: $i,X: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) ) ) )
= $true ),
inference(copy,[status(thm)],[235]) ).
thf(357,plain,
( ( ! [Y: $i] :
( ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) )
= ( inverse @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[234]) ).
thf(358,plain,
( ( ! [Z: $i] :
( ( domain_of @ ( inverse @ Z ) )
= ( range_of @ Z ) ) )
= $true ),
inference(copy,[status(thm)],[233]) ).
thf(359,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ Z @ X @ ( singleton @ Y ) ) @ null_class ) )
= ( domain @ Z @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[232]) ).
thf(360,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ Z @ ( singleton @ X ) @ Y ) @ null_class ) )
= ( range @ Z @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[231]) ).
thf(361,plain,
( ( ! [Xr: $i,X: $i] :
( ( range_of @ ( restrict @ Xr @ X @ universal_class ) )
= ( image @ Xr @ X ) ) )
= $true ),
inference(copy,[status(thm)],[230]) ).
thf(362,plain,
( ( ! [X: $i] :
( ( union @ X @ ( singleton @ X ) )
= ( successor @ X ) ) )
= $true ),
inference(copy,[status(thm)],[229]) ).
thf(363,plain,
( ( subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[228]) ).
thf(364,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ successor_relation )
| ( ( successor @ X )
= Y ) ) )
= $true ),
inference(copy,[status(thm)],[227]) ).
thf(365,plain,
( ( ! [X: $i,Y: $i] :
( ( ( successor @ X )
!= Y )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ X @ Y ) @ successor_relation ) ) )
= $true ),
inference(copy,[status(thm)],[303]) ).
thf(366,plain,
( ( ! [X: $i] :
( ~ ( inductive @ X )
| ( member @ null_class @ X ) ) )
= $true ),
inference(copy,[status(thm)],[225]) ).
thf(367,plain,
( ( ! [X: $i] :
( ~ ( inductive @ X )
| ( subclass @ ( image @ successor_relation @ X ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[224]) ).
thf(368,plain,
( ( ! [X: $i] :
( ~ ( member @ null_class @ X )
| ~ ( subclass @ ( image @ successor_relation @ X ) @ X )
| ( inductive @ X ) ) )
= $true ),
inference(copy,[status(thm)],[223]) ).
thf(369,plain,
( ( inductive @ omega )
= $true ),
inference(copy,[status(thm)],[222]) ).
thf(370,plain,
( ( ! [Y: $i] :
( ~ ( inductive @ Y )
| ( subclass @ omega @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[221]) ).
thf(371,plain,
( ( member @ omega @ universal_class )
= $true ),
inference(copy,[status(thm)],[220]) ).
thf(372,plain,
( ( ! [X: $i] :
( ( domain_of @ ( restrict @ element_relation @ universal_class @ X ) )
= ( sum_class @ X ) ) )
= $true ),
inference(copy,[status(thm)],[219]) ).
thf(373,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ ( sum_class @ X ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[218]) ).
thf(374,plain,
( ( ! [X: $i] :
( ( complement @ ( image @ element_relation @ ( complement @ X ) ) )
= ( power_class @ X ) ) )
= $true ),
inference(copy,[status(thm)],[217]) ).
thf(375,plain,
( ( ! [U: $i] :
( ~ ( member @ U @ universal_class )
| ( member @ ( power_class @ U ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[216]) ).
thf(376,plain,
( ( ! [Yr: $i,Xr: $i] : ( subclass @ ( compose @ Yr @ Xr ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[215]) ).
thf(377,plain,
( ( ! [Y: $i,Z: $i,Yr: $i,Xr: $i] :
( ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( compose @ Yr @ Xr ) )
| ( member @ Z @ ( image @ Yr @ ( image @ Xr @ ( singleton @ Y ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[214]) ).
thf(378,plain,
( ( ! [Z: $i,Yr: $i,Xr: $i,Y: $i] :
( ~ ( member @ Z @ ( image @ Yr @ ( image @ Xr @ ( singleton @ Y ) ) ) )
| ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ Y @ Z ) @ ( compose @ Yr @ Xr ) ) ) )
= $true ),
inference(copy,[status(thm)],[213]) ).
thf(379,plain,
( ( ! [X: $i] :
( ~ ( single_valued_class @ X )
| ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation ) ) )
= $true ),
inference(copy,[status(thm)],[212]) ).
thf(380,plain,
( ( ! [X: $i] :
( ~ ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation )
| ( single_valued_class @ X ) ) )
= $true ),
inference(copy,[status(thm)],[211]) ).
thf(381,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) ) ) )
= $true ),
inference(copy,[status(thm)],[210]) ).
thf(382,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation ) ) )
= $true ),
inference(copy,[status(thm)],[209]) ).
thf(383,plain,
( ( ! [Xf: $i] :
( ~ ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation )
| ( function @ Xf ) ) )
= $true ),
inference(copy,[status(thm)],[208]) ).
thf(384,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ ( image @ Xf @ X ) @ universal_class ) ) ) )
= $true ),
inference(copy,[status(thm)],[302]) ).
thf(385,plain,
( ( ! [X: $i] :
( ( X = null_class )
| ( member @ ( regular @ X ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[206]) ).
thf(386,plain,
( ( ! [X: $i] :
( ( X = null_class )
| ( ( intersection @ X @ ( regular @ X ) )
= null_class ) ) )
= $true ),
inference(copy,[status(thm)],[205]) ).
thf(387,plain,
( ( ! [Xf: $i,Y: $i] :
( ( sum_class @ ( image @ Xf @ ( singleton @ Y ) ) )
= ( apply @ Xf @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[204]) ).
thf(388,plain,
( ( function @ choice )
= $true ),
inference(copy,[status(thm)],[203]) ).
thf(389,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( Y = null_class )
| ( member @ ( apply @ choice @ Y ) @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[202]) ).
thf(390,plain,
( ( ! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ Xf ) ) )
= $true ),
inference(copy,[status(thm)],[201]) ).
thf(391,plain,
( ( ! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ ( inverse @ Xf ) ) ) )
= $true ),
inference(copy,[status(thm)],[200]) ).
thf(392,plain,
( ( ! [Xf: $i] :
( ~ ( function @ ( inverse @ Xf ) )
| ~ ( function @ Xf )
| ( one_to_one @ Xf ) ) )
= $true ),
inference(copy,[status(thm)],[199]) ).
thf(393,plain,
( ( ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( complement @ ( compose @ ( complement @ element_relation ) @ ( inverse @ element_relation ) ) ) ) )
= subset_relation )
= $true ),
inference(copy,[status(thm)],[198]) ).
thf(394,plain,
( ( ( intersection @ ( inverse @ subset_relation ) @ subset_relation )
= identity_relation )
= $true ),
inference(copy,[status(thm)],[197]) ).
thf(395,plain,
( ( ! [Xr: $i] :
( ( complement @ ( domain_of @ ( intersection @ Xr @ identity_relation ) ) )
= ( diagonalise @ Xr ) ) )
= $true ),
inference(copy,[status(thm)],[196]) ).
thf(396,plain,
( ( ! [X: $i] :
( ( intersection @ ( domain_of @ X ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ X ) ) )
= ( cantor @ X ) ) )
= $true ),
inference(copy,[status(thm)],[195]) ).
thf(397,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( function @ Xf ) ) )
= $true ),
inference(copy,[status(thm)],[194]) ).
thf(398,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( ( cross_product @ ( domain_of @ ( domain_of @ Xf ) ) @ ( domain_of @ ( domain_of @ Xf ) ) )
= ( domain_of @ Xf ) ) ) )
= $true ),
inference(copy,[status(thm)],[193]) ).
thf(399,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[192]) ).
thf(400,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( ( cross_product @ ( domain_of @ ( domain_of @ Xf ) ) @ ( domain_of @ ( domain_of @ Xf ) ) )
!= ( domain_of @ Xf ) )
| ~ ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) )
| ( operation @ Xf ) ) )
= $true ),
inference(copy,[status(thm)],[301]) ).
thf(401,plain,
( ( ! [Xh: $i] :
( ! [Xf1: $i,Xf2: $i] :
~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( function @ Xh ) ) )
= $true ),
inference(copy,[status(thm)],[300]) ).
thf(402,plain,
( ( ! [Xh: $i,Xf1: $i] :
( ! [Xf2: $i] :
~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( domain_of @ ( domain_of @ Xf1 ) )
= ( domain_of @ Xh ) ) ) )
= $true ),
inference(copy,[status(thm)],[299]) ).
thf(403,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( subclass @ ( range_of @ Xh ) @ ( domain_of @ ( domain_of @ Xf2 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[188]) ).
thf(404,plain,
( ( ! [Xh: $i] :
( ~ ( function @ Xh )
| ! [Xf1: $i] :
( ( ( domain_of @ ( domain_of @ Xf1 ) )
!= ( domain_of @ Xh ) )
| ! [Xf2: $i] :
( ~ ( subclass @ ( range_of @ Xh ) @ ( domain_of @ ( domain_of @ Xf2 ) ) )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[298]) ).
thf(405,plain,
( ( ! [Xh: $i,Xf1: $i] :
( ! [Xf2: $i] :
~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf1 ) ) )
= $true ),
inference(copy,[status(thm)],[297]) ).
thf(406,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf2 ) ) )
= $true ),
inference(copy,[status(thm)],[185]) ).
thf(407,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ) )
= $true ),
inference(copy,[status(thm)],[184]) ).
thf(408,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ ( domain_of @ Xf1 ) )
| ( ( apply @ Xf2 @ ( ordered_pair @ ( apply @ Xh @ X ) @ ( apply @ Xh @ Y ) ) )
= ( apply @ Xh @ ( apply @ Xf1 @ ( ordered_pair @ X @ Y ) ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[296]) ).
thf(409,plain,
( ( ! [Xf1: $i] :
( ~ ( operation @ Xf1 )
| ! [Xf2: $i] :
( ~ ( operation @ Xf2 )
| ! [Xh: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) @ ( domain_of @ Xf1 ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[295]) ).
thf(410,plain,
( ( ! [Xf1: $i] :
( ~ ( operation @ Xf1 )
| ! [Xf2: $i] :
( ~ ( operation @ Xf2 )
| ! [Xh: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( apply @ Xf2 @ ( ordered_pair @ ( apply @ Xh @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) ) @ ( apply @ Xh @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) ) )
!= ( apply @ Xh @ ( apply @ Xf1 @ ( ordered_pair @ ( not_homomorphism1 @ Xh @ Xf1 @ Xf2 ) @ ( not_homomorphism2 @ Xh @ Xf1 @ Xf2 ) ) ) ) )
| ( homomorphism @ Xh @ Xf1 @ Xf2 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[294]) ).
thf(411,plain,
( ( ! [X: $i,Y: $i] :
( ! [U: $i,V: $i] :
~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( member @ X @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[293]) ).
thf(412,plain,
( ( ! [X: $i,Y: $i] :
( ! [U: $i,V: $i] :
~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( member @ Y @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[292]) ).
thf(413,plain,
( ( ! [U: $i,V: $i] :
( ! [X: $i,Y: $i] :
~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ U @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[291]) ).
thf(414,plain,
( ( ! [U: $i,V: $i] :
( ! [X: $i,Y: $i] :
~ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) )
| ( member @ V @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[290]) ).
thf(415,plain,
( ( ! [X: $i] : ( subclass @ X @ X ) )
= $true ),
inference(copy,[status(thm)],[176]) ).
thf(416,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ! [Z: $i] :
( ~ ( subclass @ Y @ Z )
| ( subclass @ X @ Z ) ) ) )
= $true ),
inference(copy,[status(thm)],[289]) ).
thf(417,plain,
( ( ! [X: $i,Y: $i] :
( ( X = Y )
| ( member @ ( not_subclass_element @ X @ Y ) @ X )
| ( member @ ( not_subclass_element @ Y @ X ) @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[174]) ).
thf(418,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ( X = Y )
| ( member @ ( not_subclass_element @ Y @ X ) @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[173]) ).
thf(419,plain,
( ( ! [Y: $i,X: $i] :
( ~ ( member @ ( not_subclass_element @ Y @ X ) @ X )
| ( X = Y )
| ( member @ ( not_subclass_element @ X @ Y ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[172]) ).
thf(420,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ~ ( member @ ( not_subclass_element @ Y @ X ) @ X )
| ( X = Y ) ) )
= $true ),
inference(copy,[status(thm)],[171]) ).
thf(421,plain,
( ( ! [Y: $i,X: $i] :
~ ( member @ Y @ ( intersection @ ( complement @ X ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[170]) ).
thf(422,plain,
( ( ! [Z: $i] :
~ ( member @ Z @ null_class ) )
= $true ),
inference(copy,[status(thm)],[169]) ).
thf(423,plain,
( ( ! [X: $i] : ( subclass @ null_class @ X ) )
= $true ),
inference(copy,[status(thm)],[168]) ).
thf(424,plain,
( ( ! [X: $i] :
( ~ ( subclass @ X @ null_class )
| ( X = null_class ) ) )
= $true ),
inference(copy,[status(thm)],[167]) ).
thf(425,plain,
( ( ! [Z: $i] :
( ( Z = null_class )
| ( member @ ( not_subclass_element @ Z @ null_class ) @ Z ) ) )
= $true ),
inference(copy,[status(thm)],[166]) ).
thf(426,plain,
( ( member @ null_class @ universal_class )
= $true ),
inference(copy,[status(thm)],[165]) ).
thf(427,plain,
( ( ! [X: $i,Y: $i] :
( ( unordered_pair @ X @ Y )
= ( unordered_pair @ Y @ X ) ) )
= $true ),
inference(copy,[status(thm)],[164]) ).
thf(428,plain,
( ( ! [X: $i,Y: $i] : ( subclass @ ( singleton @ X ) @ ( unordered_pair @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[163]) ).
thf(429,plain,
( ( ! [Y: $i,X: $i] : ( subclass @ ( singleton @ Y ) @ ( unordered_pair @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[162]) ).
thf(430,plain,
( ( ! [Y: $i] :
( ! [X: $i] :
( ( unordered_pair @ X @ Y )
= ( singleton @ X ) )
| ( member @ Y @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[288]) ).
thf(431,plain,
( ( ! [X: $i] :
( ! [Y: $i] :
( ( unordered_pair @ X @ Y )
= ( singleton @ Y ) )
| ( member @ X @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[287]) ).
thf(432,plain,
( ( ! [X: $i,Y: $i] :
( ( ( unordered_pair @ X @ Y )
= null_class )
| ( member @ X @ universal_class )
| ( member @ Y @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[159]) ).
thf(433,plain,
( ( ! [X: $i,Y: $i,Z: $i] :
( ( ( unordered_pair @ X @ Y )
!= ( unordered_pair @ X @ Z ) )
| ~ ( member @ ( ordered_pair @ Y @ Z ) @ ( cross_product @ universal_class @ universal_class ) )
| ( Y = Z ) ) )
= $true ),
inference(copy,[status(thm)],[286]) ).
thf(434,plain,
( ( ! [X: $i,Z: $i,Y: $i] :
( ( ( unordered_pair @ X @ Z )
!= ( unordered_pair @ Y @ Z ) )
| ~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ universal_class @ universal_class ) )
| ( X = Y ) ) )
= $true ),
inference(copy,[status(thm)],[285]) ).
thf(435,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ! [Y: $i] :
( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(copy,[status(thm)],[284]) ).
thf(436,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ! [X: $i] :
( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(copy,[status(thm)],[283]) ).
thf(437,plain,
( ( ! [X: $i,Y: $i] :
( ! [U: $i,V: $i] :
~ ( member @ ( ordered_pair @ X @ Y ) @ ( cross_product @ U @ V ) )
| ( ( unordered_pair @ X @ Y )
!= null_class ) ) )
= $true ),
inference(copy,[status(thm)],[282]) ).
thf(438,plain,
( ( ! [X: $i,Z: $i] :
( ~ ( member @ X @ Z )
| ! [Y: $i] :
( ~ ( member @ Y @ Z )
| ( subclass @ ( unordered_pair @ X @ Y ) @ Z ) ) ) )
= $true ),
inference(copy,[status(thm)],[281]) ).
thf(439,plain,
( ( ! [X: $i] : ( member @ ( singleton @ X ) @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[152]) ).
thf(440,plain,
( ( ! [Y: $i,X: $i] : ( member @ ( singleton @ Y ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[151]) ).
thf(441,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ X @ ( singleton @ X ) ) ) )
= $true ),
inference(copy,[status(thm)],[150]) ).
thf(442,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( ( singleton @ X )
!= null_class ) ) )
= $true ),
inference(copy,[status(thm)],[280]) ).
thf(443,plain,
( ( member @ null_class @ ( singleton @ null_class ) )
= $true ),
inference(copy,[status(thm)],[148]) ).
thf(444,plain,
( ( ! [Y: $i,X: $i] :
( ~ ( member @ Y @ ( singleton @ X ) )
| ( Y = X ) ) )
= $true ),
inference(copy,[status(thm)],[147]) ).
thf(445,plain,
( ( ! [X: $i] :
( ( ( singleton @ X )
= null_class )
| ( member @ X @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[279]) ).
thf(446,plain,
( ( ! [X: $i,Y: $i] :
( ( ( singleton @ X )
!= ( singleton @ Y ) )
| ~ ( member @ X @ universal_class )
| ( X = Y ) ) )
= $true ),
inference(copy,[status(thm)],[278]) ).
thf(447,plain,
( ( ! [X: $i,Y: $i] :
( ( ( singleton @ X )
!= ( singleton @ Y ) )
| ~ ( member @ Y @ universal_class )
| ( X = Y ) ) )
= $true ),
inference(copy,[status(thm)],[277]) ).
thf(448,plain,
( ( ! [Y: $i,Z: $i,X: $i] :
( ( ( unordered_pair @ Y @ Z )
!= ( singleton @ X ) )
| ~ ( member @ X @ universal_class )
| ( X = Y )
| ( X = Z ) ) )
= $true ),
inference(copy,[status(thm)],[276]) ).
thf(449,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( member @ ( member_of @ ( singleton @ Y ) ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[142]) ).
thf(450,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( ( singleton @ ( member_of @ ( singleton @ Y ) ) )
= ( singleton @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[141]) ).
thf(451,plain,
( ( ! [X: $i] :
( ( ( member_of @ X )
= X )
| ( member @ ( member_of @ X ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[275]) ).
thf(452,plain,
( ( ! [X: $i] :
( ( ( singleton @ ( member_of @ X ) )
= X )
| ( ( member_of @ X )
= X ) ) )
= $true ),
inference(copy,[status(thm)],[139]) ).
thf(453,plain,
( ( ~ $false ) = $true ),
inference(copy,[status(thm)],[274]) ).
thf(454,plain,
( ( ( member_of @ ( singleton @ u ) )
= u )
= $false ),
inference(extcnf_not_pos,[status(thm)],[319]) ).
thf(455,plain,
! [SV1: $i] :
( ( ! [SY258: $i] :
( ~ ( subclass @ SV1 @ SY258 )
| ! [SY259: $i] :
( ~ ( member @ SY259 @ SV1 )
| ( member @ SY259 @ SY258 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[320]) ).
thf(456,plain,
! [SV2: $i] :
( ( ! [SY260: $i] :
( ( member @ ( not_subclass_element @ SV2 @ SY260 ) @ SV2 )
| ( subclass @ SV2 @ SY260 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[321]) ).
thf(457,plain,
! [SV3: $i] :
( ( ! [SY261: $i] :
( ~ ( member @ ( not_subclass_element @ SV3 @ SY261 ) @ SY261 )
| ( subclass @ SV3 @ SY261 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[322]) ).
thf(458,plain,
! [SV4: $i] :
( ( subclass @ SV4 @ universal_class )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[323]) ).
thf(459,plain,
! [SV5: $i] :
( ( ! [SY262: $i] :
( ( SV5 != SY262 )
| ( subclass @ SV5 @ SY262 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[324]) ).
thf(460,plain,
! [SV6: $i] :
( ( ! [SY263: $i] :
( ( SV6 != SY263 )
| ( subclass @ SY263 @ SV6 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[325]) ).
thf(461,plain,
! [SV7: $i] :
( ( ! [SY264: $i] :
( ~ ( subclass @ SV7 @ SY264 )
| ~ ( subclass @ SY264 @ SV7 )
| ( SV7 = SY264 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[326]) ).
thf(462,plain,
! [SV8: $i] :
( ( ! [SY265: $i,SY266: $i] :
( ~ ( member @ SV8 @ ( unordered_pair @ SY265 @ SY266 ) )
| ( SV8 = SY265 )
| ( SV8 = SY266 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[327]) ).
thf(463,plain,
! [SV9: $i] :
( ( ~ ( member @ SV9 @ universal_class )
| ! [SY267: $i] : ( member @ SV9 @ ( unordered_pair @ SV9 @ SY267 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[328]) ).
thf(464,plain,
! [SV10: $i] :
( ( ~ ( member @ SV10 @ universal_class )
| ! [SY268: $i] : ( member @ SV10 @ ( unordered_pair @ SY268 @ SV10 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[329]) ).
thf(465,plain,
! [SV11: $i] :
( ( ! [SY269: $i] : ( member @ ( unordered_pair @ SV11 @ SY269 ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[330]) ).
thf(466,plain,
! [SV12: $i] :
( ( ( unordered_pair @ SV12 @ SV12 )
= ( singleton @ SV12 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[331]) ).
thf(467,plain,
! [SV13: $i] :
( ( ! [SY270: $i] :
( ( unordered_pair @ ( singleton @ SV13 ) @ ( unordered_pair @ SV13 @ ( singleton @ SY270 ) ) )
= ( ordered_pair @ SV13 @ SY270 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[332]) ).
thf(468,plain,
! [SV14: $i] :
( ( ! [SY271: $i,SY272: $i] :
( ! [SY273: $i] :
~ ( member @ ( ordered_pair @ SV14 @ SY271 ) @ ( cross_product @ SY272 @ SY273 ) )
| ( member @ SV14 @ SY272 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[333]) ).
thf(469,plain,
! [SV15: $i] :
( ( ! [SY274: $i,SY275: $i,SY276: $i] :
( ~ ( member @ ( ordered_pair @ SV15 @ SY274 ) @ ( cross_product @ SY275 @ SY276 ) )
| ( member @ SY274 @ SY276 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[334]) ).
thf(470,plain,
! [SV16: $i] :
( ( ! [SY277: $i] :
( ~ ( member @ SV16 @ SY277 )
| ! [SY278: $i,SY279: $i] :
( ~ ( member @ SY278 @ SY279 )
| ( member @ ( ordered_pair @ SV16 @ SY278 ) @ ( cross_product @ SY277 @ SY279 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[335]) ).
thf(471,plain,
! [SV17: $i] :
( ( ! [SY280: $i,SY281: $i] :
~ ( member @ SV17 @ ( cross_product @ SY280 @ SY281 ) )
| ( ( ordered_pair @ ( first @ SV17 ) @ ( second @ SV17 ) )
= SV17 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[336]) ).
thf(472,plain,
! [SV18: $i] :
( ( ! [SY282: $i] :
( ~ ( member @ ( ordered_pair @ SV18 @ SY282 ) @ element_relation )
| ( member @ SV18 @ SY282 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[338]) ).
thf(473,plain,
! [SV19: $i] :
( ( ! [SY283: $i] :
( ~ ( member @ ( ordered_pair @ SV19 @ SY283 ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ SV19 @ SY283 )
| ( member @ ( ordered_pair @ SV19 @ SY283 ) @ element_relation ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[339]) ).
thf(474,plain,
! [SV20: $i] :
( ( ! [SY284: $i] :
( ! [SY285: $i] :
~ ( member @ SV20 @ ( intersection @ SY284 @ SY285 ) )
| ( member @ SV20 @ SY284 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[340]) ).
thf(475,plain,
! [SV21: $i] :
( ( ! [SY286: $i,SY287: $i] :
( ~ ( member @ SV21 @ ( intersection @ SY286 @ SY287 ) )
| ( member @ SV21 @ SY287 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[341]) ).
thf(476,plain,
! [SV22: $i] :
( ( ! [SY288: $i] :
( ~ ( member @ SV22 @ SY288 )
| ! [SY289: $i] :
( ~ ( member @ SV22 @ SY289 )
| ( member @ SV22 @ ( intersection @ SY288 @ SY289 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[342]) ).
thf(477,plain,
! [SV23: $i] :
( ( ! [SY290: $i] :
( ~ ( member @ SV23 @ ( complement @ SY290 ) )
| ~ ( member @ SV23 @ SY290 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[343]) ).
thf(478,plain,
! [SV24: $i] :
( ( ~ ( member @ SV24 @ universal_class )
| ! [SY291: $i] :
( ( member @ SV24 @ ( complement @ SY291 ) )
| ( member @ SV24 @ SY291 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[344]) ).
thf(479,plain,
! [SV25: $i] :
( ( ! [SY292: $i] :
( ( complement @ ( intersection @ ( complement @ SV25 ) @ ( complement @ SY292 ) ) )
= ( union @ SV25 @ SY292 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[345]) ).
thf(480,plain,
! [SV26: $i] :
( ( ! [SY293: $i] :
( ( intersection @ ( complement @ ( intersection @ SV26 @ SY293 ) ) @ ( complement @ ( intersection @ ( complement @ SV26 ) @ ( complement @ SY293 ) ) ) )
= ( symmetric_difference @ SV26 @ SY293 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[346]) ).
thf(481,plain,
! [SV27: $i] :
( ( ! [SY294: $i,SY295: $i] :
( ( intersection @ SV27 @ ( cross_product @ SY294 @ SY295 ) )
= ( restrict @ SV27 @ SY294 @ SY295 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[347]) ).
thf(482,plain,
! [SV28: $i] :
( ( ! [SY296: $i,SY297: $i] :
( ( intersection @ ( cross_product @ SV28 @ SY296 ) @ SY297 )
= ( restrict @ SY297 @ SV28 @ SY296 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[348]) ).
thf(483,plain,
! [SV29: $i] :
( ( ! [SY298: $i] :
( ( ( restrict @ SV29 @ ( singleton @ SY298 ) @ universal_class )
!= null_class )
| ~ ( member @ SY298 @ ( domain_of @ SV29 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[349]) ).
thf(484,plain,
! [SV30: $i] :
( ( ~ ( member @ SV30 @ universal_class )
| ! [SY299: $i] :
( ( ( restrict @ SY299 @ ( singleton @ SV30 ) @ universal_class )
= null_class )
| ( member @ SV30 @ ( domain_of @ SY299 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[350]) ).
thf(485,plain,
! [SV31: $i] :
( ( subclass @ ( rotate @ SV31 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[351]) ).
thf(486,plain,
! [SV32: $i] :
( ( ! [SY300: $i,SY301: $i,SY302: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SY300 ) @ SY301 ) @ ( rotate @ SY302 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY300 @ SY301 ) @ SV32 ) @ SY302 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[352]) ).
thf(487,plain,
! [SV33: $i] :
( ( ! [SY303: $i,SY304: $i,SY305: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SY303 ) @ SY304 ) @ SY305 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SY304 @ SV33 ) @ SY303 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY304 @ SV33 ) @ SY303 ) @ ( rotate @ SY305 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[353]) ).
thf(488,plain,
! [SV34: $i] :
( ( subclass @ ( flip @ SV34 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[354]) ).
thf(489,plain,
! [SV35: $i] :
( ( ! [SY306: $i,SY307: $i,SY308: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV35 @ SY306 ) @ SY307 ) @ ( flip @ SY308 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY306 @ SV35 ) @ SY307 ) @ SY308 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[355]) ).
thf(490,plain,
! [SV36: $i] :
( ( ! [SY309: $i,SY310: $i,SY311: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV36 @ SY309 ) @ SY310 ) @ SY311 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SY309 @ SV36 ) @ SY310 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY309 @ SV36 ) @ SY310 ) @ ( flip @ SY311 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[356]) ).
thf(491,plain,
! [SV37: $i] :
( ( ( domain_of @ ( flip @ ( cross_product @ SV37 @ universal_class ) ) )
= ( inverse @ SV37 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[357]) ).
thf(492,plain,
! [SV38: $i] :
( ( ( domain_of @ ( inverse @ SV38 ) )
= ( range_of @ SV38 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[358]) ).
thf(493,plain,
! [SV39: $i] :
( ( ! [SY312: $i,SY313: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ SV39 @ SY312 @ ( singleton @ SY313 ) ) @ null_class ) )
= ( domain @ SV39 @ SY312 @ SY313 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[359]) ).
thf(494,plain,
! [SV40: $i] :
( ( ! [SY314: $i,SY315: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ SV40 @ ( singleton @ SY314 ) @ SY315 ) @ null_class ) )
= ( range @ SV40 @ SY314 @ SY315 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[360]) ).
thf(495,plain,
! [SV41: $i] :
( ( ! [SY316: $i] :
( ( range_of @ ( restrict @ SV41 @ SY316 @ universal_class ) )
= ( image @ SV41 @ SY316 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[361]) ).
thf(496,plain,
! [SV42: $i] :
( ( ( union @ SV42 @ ( singleton @ SV42 ) )
= ( successor @ SV42 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[362]) ).
thf(497,plain,
! [SV43: $i] :
( ( ! [SY317: $i] :
( ~ ( member @ ( ordered_pair @ SV43 @ SY317 ) @ successor_relation )
| ( ( successor @ SV43 )
= SY317 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[364]) ).
thf(498,plain,
! [SV44: $i] :
( ( ! [SY318: $i] :
( ( ( successor @ SV44 )
!= SY318 )
| ~ ( member @ ( ordered_pair @ SV44 @ SY318 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV44 @ SY318 ) @ successor_relation ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[365]) ).
thf(499,plain,
! [SV45: $i] :
( ( ~ ( inductive @ SV45 )
| ( member @ null_class @ SV45 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[366]) ).
thf(500,plain,
! [SV46: $i] :
( ( ~ ( inductive @ SV46 )
| ( subclass @ ( image @ successor_relation @ SV46 ) @ SV46 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[367]) ).
thf(501,plain,
! [SV47: $i] :
( ( ~ ( member @ null_class @ SV47 )
| ~ ( subclass @ ( image @ successor_relation @ SV47 ) @ SV47 )
| ( inductive @ SV47 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[368]) ).
thf(502,plain,
! [SV48: $i] :
( ( ~ ( inductive @ SV48 )
| ( subclass @ omega @ SV48 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[370]) ).
thf(503,plain,
! [SV49: $i] :
( ( ( domain_of @ ( restrict @ element_relation @ universal_class @ SV49 ) )
= ( sum_class @ SV49 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[372]) ).
thf(504,plain,
! [SV50: $i] :
( ( ~ ( member @ SV50 @ universal_class )
| ( member @ ( sum_class @ SV50 ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[373]) ).
thf(505,plain,
! [SV51: $i] :
( ( ( complement @ ( image @ element_relation @ ( complement @ SV51 ) ) )
= ( power_class @ SV51 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[374]) ).
thf(506,plain,
! [SV52: $i] :
( ( ~ ( member @ SV52 @ universal_class )
| ( member @ ( power_class @ SV52 ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[375]) ).
thf(507,plain,
! [SV53: $i] :
( ( ! [SY319: $i] : ( subclass @ ( compose @ SV53 @ SY319 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[376]) ).
thf(508,plain,
! [SV54: $i] :
( ( ! [SY320: $i,SY321: $i,SY322: $i] :
( ~ ( member @ ( ordered_pair @ SV54 @ SY320 ) @ ( compose @ SY321 @ SY322 ) )
| ( member @ SY320 @ ( image @ SY321 @ ( image @ SY322 @ ( singleton @ SV54 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[377]) ).
thf(509,plain,
! [SV55: $i] :
( ( ! [SY323: $i,SY324: $i,SY325: $i] :
( ~ ( member @ SV55 @ ( image @ SY323 @ ( image @ SY324 @ ( singleton @ SY325 ) ) ) )
| ~ ( member @ ( ordered_pair @ SY325 @ SV55 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SY325 @ SV55 ) @ ( compose @ SY323 @ SY324 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[378]) ).
thf(510,plain,
! [SV56: $i] :
( ( ~ ( single_valued_class @ SV56 )
| ( subclass @ ( compose @ SV56 @ ( inverse @ SV56 ) ) @ identity_relation ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[379]) ).
thf(511,plain,
! [SV57: $i] :
( ( ~ ( subclass @ ( compose @ SV57 @ ( inverse @ SV57 ) ) @ identity_relation )
| ( single_valued_class @ SV57 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[380]) ).
thf(512,plain,
! [SV58: $i] :
( ( ~ ( function @ SV58 )
| ( subclass @ SV58 @ ( cross_product @ universal_class @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[381]) ).
thf(513,plain,
! [SV59: $i] :
( ( ~ ( function @ SV59 )
| ( subclass @ ( compose @ SV59 @ ( inverse @ SV59 ) ) @ identity_relation ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[382]) ).
thf(514,plain,
! [SV60: $i] :
( ( ~ ( subclass @ SV60 @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( subclass @ ( compose @ SV60 @ ( inverse @ SV60 ) ) @ identity_relation )
| ( function @ SV60 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[383]) ).
thf(515,plain,
! [SV61: $i] :
( ( ~ ( function @ SV61 )
| ! [SY326: $i] :
( ~ ( member @ SY326 @ universal_class )
| ( member @ ( image @ SV61 @ SY326 ) @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[384]) ).
thf(516,plain,
! [SV62: $i] :
( ( ( SV62 = null_class )
| ( member @ ( regular @ SV62 ) @ SV62 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[385]) ).
thf(517,plain,
! [SV63: $i] :
( ( ( SV63 = null_class )
| ( ( intersection @ SV63 @ ( regular @ SV63 ) )
= null_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[386]) ).
thf(518,plain,
! [SV64: $i] :
( ( ! [SY327: $i] :
( ( sum_class @ ( image @ SV64 @ ( singleton @ SY327 ) ) )
= ( apply @ SV64 @ SY327 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[387]) ).
thf(519,plain,
! [SV65: $i] :
( ( ~ ( member @ SV65 @ universal_class )
| ( SV65 = null_class )
| ( member @ ( apply @ choice @ SV65 ) @ SV65 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[389]) ).
thf(520,plain,
! [SV66: $i] :
( ( ~ ( one_to_one @ SV66 )
| ( function @ SV66 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[390]) ).
thf(521,plain,
! [SV67: $i] :
( ( ~ ( one_to_one @ SV67 )
| ( function @ ( inverse @ SV67 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[391]) ).
thf(522,plain,
! [SV68: $i] :
( ( ~ ( function @ ( inverse @ SV68 ) )
| ~ ( function @ SV68 )
| ( one_to_one @ SV68 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[392]) ).
thf(523,plain,
! [SV69: $i] :
( ( ( complement @ ( domain_of @ ( intersection @ SV69 @ identity_relation ) ) )
= ( diagonalise @ SV69 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[395]) ).
thf(524,plain,
! [SV70: $i] :
( ( ( intersection @ ( domain_of @ SV70 ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ SV70 ) ) )
= ( cantor @ SV70 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[396]) ).
thf(525,plain,
! [SV71: $i] :
( ( ~ ( operation @ SV71 )
| ( function @ SV71 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[397]) ).
thf(526,plain,
! [SV72: $i] :
( ( ~ ( operation @ SV72 )
| ( ( cross_product @ ( domain_of @ ( domain_of @ SV72 ) ) @ ( domain_of @ ( domain_of @ SV72 ) ) )
= ( domain_of @ SV72 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[398]) ).
thf(527,plain,
! [SV73: $i] :
( ( ~ ( operation @ SV73 )
| ( subclass @ ( range_of @ SV73 ) @ ( domain_of @ ( domain_of @ SV73 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[399]) ).
thf(528,plain,
! [SV74: $i] :
( ( ~ ( function @ SV74 )
| ( ( cross_product @ ( domain_of @ ( domain_of @ SV74 ) ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
!= ( domain_of @ SV74 ) )
| ~ ( subclass @ ( range_of @ SV74 ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
| ( operation @ SV74 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[400]) ).
thf(529,plain,
! [SV75: $i] :
( ( ! [SY328: $i,SY329: $i] :
~ ( compatible @ SV75 @ SY328 @ SY329 )
| ( function @ SV75 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[401]) ).
thf(530,plain,
! [SV76: $i] :
( ( ! [SY330: $i] :
( ! [SY331: $i] :
~ ( compatible @ SV76 @ SY330 @ SY331 )
| ( ( domain_of @ ( domain_of @ SY330 ) )
= ( domain_of @ SV76 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[402]) ).
thf(531,plain,
! [SV77: $i] :
( ( ! [SY332: $i,SY333: $i] :
( ~ ( compatible @ SV77 @ SY332 @ SY333 )
| ( subclass @ ( range_of @ SV77 ) @ ( domain_of @ ( domain_of @ SY333 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[403]) ).
thf(532,plain,
! [SV78: $i] :
( ( ~ ( function @ SV78 )
| ! [SY334: $i] :
( ( ( domain_of @ ( domain_of @ SY334 ) )
!= ( domain_of @ SV78 ) )
| ! [SY335: $i] :
( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SY335 ) ) )
| ( compatible @ SV78 @ SY334 @ SY335 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[404]) ).
thf(533,plain,
! [SV79: $i] :
( ( ! [SY336: $i] :
( ! [SY337: $i] :
~ ( homomorphism @ SV79 @ SY336 @ SY337 )
| ( operation @ SY336 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[405]) ).
thf(534,plain,
! [SV80: $i] :
( ( ! [SY338: $i,SY339: $i] :
( ~ ( homomorphism @ SV80 @ SY338 @ SY339 )
| ( operation @ SY339 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[406]) ).
thf(535,plain,
! [SV81: $i] :
( ( ! [SY340: $i,SY341: $i] :
( ~ ( homomorphism @ SV81 @ SY340 @ SY341 )
| ( compatible @ SV81 @ SY340 @ SY341 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[407]) ).
thf(536,plain,
! [SV82: $i] :
( ( ! [SY342: $i,SY343: $i] :
( ~ ( homomorphism @ SV82 @ SY342 @ SY343 )
| ! [SY344: $i,SY345: $i] :
( ~ ( member @ ( ordered_pair @ SY344 @ SY345 ) @ ( domain_of @ SY342 ) )
| ( ( apply @ SY343 @ ( ordered_pair @ ( apply @ SV82 @ SY344 ) @ ( apply @ SV82 @ SY345 ) ) )
= ( apply @ SV82 @ ( apply @ SY342 @ ( ordered_pair @ SY344 @ SY345 ) ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[408]) ).
thf(537,plain,
! [SV83: $i] :
( ( ~ ( operation @ SV83 )
| ! [SY346: $i] :
( ~ ( operation @ SY346 )
| ! [SY347: $i] :
( ~ ( compatible @ SY347 @ SV83 @ SY346 )
| ( homomorphism @ SY347 @ SV83 @ SY346 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SY347 @ SV83 @ SY346 ) @ ( not_homomorphism2 @ SY347 @ SV83 @ SY346 ) ) @ ( domain_of @ SV83 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[409]) ).
thf(538,plain,
! [SV84: $i] :
( ( ~ ( operation @ SV84 )
| ! [SY348: $i] :
( ~ ( operation @ SY348 )
| ! [SY349: $i] :
( ~ ( compatible @ SY349 @ SV84 @ SY348 )
| ( ( apply @ SY348 @ ( ordered_pair @ ( apply @ SY349 @ ( not_homomorphism1 @ SY349 @ SV84 @ SY348 ) ) @ ( apply @ SY349 @ ( not_homomorphism2 @ SY349 @ SV84 @ SY348 ) ) ) )
!= ( apply @ SY349 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SY349 @ SV84 @ SY348 ) @ ( not_homomorphism2 @ SY349 @ SV84 @ SY348 ) ) ) ) )
| ( homomorphism @ SY349 @ SV84 @ SY348 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[410]) ).
thf(539,plain,
! [SV85: $i] :
( ( ! [SY350: $i] :
( ! [SY351: $i,SY352: $i] :
~ ( member @ ( ordered_pair @ SV85 @ SY350 ) @ ( cross_product @ SY351 @ SY352 ) )
| ( member @ SV85 @ ( unordered_pair @ SV85 @ SY350 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[411]) ).
thf(540,plain,
! [SV86: $i] :
( ( ! [SY353: $i] :
( ! [SY354: $i,SY355: $i] :
~ ( member @ ( ordered_pair @ SV86 @ SY353 ) @ ( cross_product @ SY354 @ SY355 ) )
| ( member @ SY353 @ ( unordered_pair @ SV86 @ SY353 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[412]) ).
thf(541,plain,
! [SV87: $i] :
( ( ! [SY356: $i] :
( ! [SY357: $i,SY358: $i] :
~ ( member @ ( ordered_pair @ SV87 @ SY356 ) @ ( cross_product @ SY357 @ SY358 ) )
| ( member @ SV87 @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[413]) ).
thf(542,plain,
! [SV88: $i] :
( ( ! [SY359: $i] :
( ! [SY360: $i,SY361: $i] :
~ ( member @ ( ordered_pair @ SV88 @ SY359 ) @ ( cross_product @ SY360 @ SY361 ) )
| ( member @ SY359 @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[414]) ).
thf(543,plain,
! [SV89: $i] :
( ( subclass @ SV89 @ SV89 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[415]) ).
thf(544,plain,
! [SV90: $i] :
( ( ! [SY362: $i] :
( ~ ( subclass @ SV90 @ SY362 )
| ! [SY363: $i] :
( ~ ( subclass @ SY362 @ SY363 )
| ( subclass @ SV90 @ SY363 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[416]) ).
thf(545,plain,
! [SV91: $i] :
( ( ! [SY364: $i] :
( ( SV91 = SY364 )
| ( member @ ( not_subclass_element @ SV91 @ SY364 ) @ SV91 )
| ( member @ ( not_subclass_element @ SY364 @ SV91 ) @ SY364 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[417]) ).
thf(546,plain,
! [SV92: $i] :
( ( ! [SY365: $i] :
( ~ ( member @ ( not_subclass_element @ SV92 @ SY365 ) @ SY365 )
| ( SV92 = SY365 )
| ( member @ ( not_subclass_element @ SY365 @ SV92 ) @ SY365 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[418]) ).
thf(547,plain,
! [SV93: $i] :
( ( ! [SY366: $i] :
( ~ ( member @ ( not_subclass_element @ SV93 @ SY366 ) @ SY366 )
| ( SY366 = SV93 )
| ( member @ ( not_subclass_element @ SY366 @ SV93 ) @ SY366 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[419]) ).
thf(548,plain,
! [SV94: $i] :
( ( ! [SY367: $i] :
( ~ ( member @ ( not_subclass_element @ SV94 @ SY367 ) @ SY367 )
| ~ ( member @ ( not_subclass_element @ SY367 @ SV94 ) @ SV94 )
| ( SV94 = SY367 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[420]) ).
thf(549,plain,
! [SV95: $i] :
( ( ! [SY368: $i] :
~ ( member @ SV95 @ ( intersection @ ( complement @ SY368 ) @ SY368 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[421]) ).
thf(550,plain,
! [SV96: $i] :
( ( ~ ( member @ SV96 @ null_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[422]) ).
thf(551,plain,
! [SV97: $i] :
( ( subclass @ null_class @ SV97 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[423]) ).
thf(552,plain,
! [SV98: $i] :
( ( ~ ( subclass @ SV98 @ null_class )
| ( SV98 = null_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[424]) ).
thf(553,plain,
! [SV99: $i] :
( ( ( SV99 = null_class )
| ( member @ ( not_subclass_element @ SV99 @ null_class ) @ SV99 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[425]) ).
thf(554,plain,
! [SV100: $i] :
( ( ! [SY369: $i] :
( ( unordered_pair @ SV100 @ SY369 )
= ( unordered_pair @ SY369 @ SV100 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[427]) ).
thf(555,plain,
! [SV101: $i] :
( ( ! [SY370: $i] : ( subclass @ ( singleton @ SV101 ) @ ( unordered_pair @ SV101 @ SY370 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[428]) ).
thf(556,plain,
! [SV102: $i] :
( ( ! [SY371: $i] : ( subclass @ ( singleton @ SV102 ) @ ( unordered_pair @ SY371 @ SV102 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[429]) ).
thf(557,plain,
! [SV103: $i] :
( ( ! [SY372: $i] :
( ( unordered_pair @ SY372 @ SV103 )
= ( singleton @ SY372 ) )
| ( member @ SV103 @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[430]) ).
thf(558,plain,
! [SV104: $i] :
( ( ! [SY373: $i] :
( ( unordered_pair @ SV104 @ SY373 )
= ( singleton @ SY373 ) )
| ( member @ SV104 @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[431]) ).
thf(559,plain,
! [SV105: $i] :
( ( ! [SY374: $i] :
( ( ( unordered_pair @ SV105 @ SY374 )
= null_class )
| ( member @ SV105 @ universal_class )
| ( member @ SY374 @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[432]) ).
thf(560,plain,
! [SV106: $i] :
( ( ! [SY375: $i,SY376: $i] :
( ( ( unordered_pair @ SV106 @ SY375 )
!= ( unordered_pair @ SV106 @ SY376 ) )
| ~ ( member @ ( ordered_pair @ SY375 @ SY376 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SY375 = SY376 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[433]) ).
thf(561,plain,
! [SV107: $i] :
( ( ! [SY377: $i,SY378: $i] :
( ( ( unordered_pair @ SV107 @ SY377 )
!= ( unordered_pair @ SY378 @ SY377 ) )
| ~ ( member @ ( ordered_pair @ SV107 @ SY378 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV107 = SY378 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[434]) ).
thf(562,plain,
! [SV108: $i] :
( ( ~ ( member @ SV108 @ universal_class )
| ! [SY379: $i] :
( ( unordered_pair @ SV108 @ SY379 )
!= null_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[435]) ).
thf(563,plain,
! [SV109: $i] :
( ( ~ ( member @ SV109 @ universal_class )
| ! [SY380: $i] :
( ( unordered_pair @ SY380 @ SV109 )
!= null_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[436]) ).
thf(564,plain,
! [SV110: $i] :
( ( ! [SY381: $i] :
( ! [SY382: $i,SY383: $i] :
~ ( member @ ( ordered_pair @ SV110 @ SY381 ) @ ( cross_product @ SY382 @ SY383 ) )
| ( ( unordered_pair @ SV110 @ SY381 )
!= null_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[437]) ).
thf(565,plain,
! [SV111: $i] :
( ( ! [SY384: $i] :
( ~ ( member @ SV111 @ SY384 )
| ! [SY385: $i] :
( ~ ( member @ SY385 @ SY384 )
| ( subclass @ ( unordered_pair @ SV111 @ SY385 ) @ SY384 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[438]) ).
thf(566,plain,
! [SV112: $i] :
( ( member @ ( singleton @ SV112 ) @ universal_class )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[439]) ).
thf(567,plain,
! [SV113: $i] :
( ( ! [SY386: $i] : ( member @ ( singleton @ SV113 ) @ ( unordered_pair @ SY386 @ ( singleton @ SV113 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[440]) ).
thf(568,plain,
! [SV114: $i] :
( ( ~ ( member @ SV114 @ universal_class )
| ( member @ SV114 @ ( singleton @ SV114 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[441]) ).
thf(569,plain,
! [SV115: $i] :
( ( ~ ( member @ SV115 @ universal_class )
| ( ( singleton @ SV115 )
!= null_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[442]) ).
thf(570,plain,
! [SV116: $i] :
( ( ! [SY387: $i] :
( ~ ( member @ SV116 @ ( singleton @ SY387 ) )
| ( SV116 = SY387 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[444]) ).
thf(571,plain,
! [SV117: $i] :
( ( ( ( singleton @ SV117 )
= null_class )
| ( member @ SV117 @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[445]) ).
thf(572,plain,
! [SV118: $i] :
( ( ! [SY388: $i] :
( ( ( singleton @ SV118 )
!= ( singleton @ SY388 ) )
| ~ ( member @ SV118 @ universal_class )
| ( SV118 = SY388 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[446]) ).
thf(573,plain,
! [SV119: $i] :
( ( ! [SY389: $i] :
( ( ( singleton @ SV119 )
!= ( singleton @ SY389 ) )
| ~ ( member @ SY389 @ universal_class )
| ( SV119 = SY389 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[447]) ).
thf(574,plain,
! [SV120: $i] :
( ( ! [SY390: $i,SY391: $i] :
( ( ( unordered_pair @ SV120 @ SY390 )
!= ( singleton @ SY391 ) )
| ~ ( member @ SY391 @ universal_class )
| ( SY391 = SV120 )
| ( SY391 = SY390 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[448]) ).
thf(575,plain,
! [SV121: $i] :
( ( ~ ( member @ SV121 @ universal_class )
| ( member @ ( member_of @ ( singleton @ SV121 ) ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[449]) ).
thf(576,plain,
! [SV122: $i] :
( ( ~ ( member @ SV122 @ universal_class )
| ( ( singleton @ ( member_of @ ( singleton @ SV122 ) ) )
= ( singleton @ SV122 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[450]) ).
thf(577,plain,
! [SV123: $i] :
( ( ( ( member_of @ SV123 )
= SV123 )
| ( member @ ( member_of @ SV123 ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[451]) ).
thf(578,plain,
! [SV124: $i] :
( ( ( ( singleton @ ( member_of @ SV124 ) )
= SV124 )
| ( ( member_of @ SV124 )
= SV124 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[452]) ).
thf(579,plain,
$false = $false,
inference(extcnf_not_pos,[status(thm)],[453]) ).
thf(580,plain,
! [SV125: $i,SV1: $i] :
( ( ~ ( subclass @ SV1 @ SV125 )
| ! [SY392: $i] :
( ~ ( member @ SY392 @ SV1 )
| ( member @ SY392 @ SV125 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[455]) ).
thf(581,plain,
! [SV126: $i,SV2: $i] :
( ( ( member @ ( not_subclass_element @ SV2 @ SV126 ) @ SV2 )
| ( subclass @ SV2 @ SV126 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[456]) ).
thf(582,plain,
! [SV127: $i,SV3: $i] :
( ( ~ ( member @ ( not_subclass_element @ SV3 @ SV127 ) @ SV127 )
| ( subclass @ SV3 @ SV127 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[457]) ).
thf(583,plain,
! [SV128: $i,SV5: $i] :
( ( ( SV5 != SV128 )
| ( subclass @ SV5 @ SV128 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[459]) ).
thf(584,plain,
! [SV129: $i,SV6: $i] :
( ( ( SV6 != SV129 )
| ( subclass @ SV129 @ SV6 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[460]) ).
thf(585,plain,
! [SV130: $i,SV7: $i] :
( ( ~ ( subclass @ SV7 @ SV130 )
| ~ ( subclass @ SV130 @ SV7 )
| ( SV7 = SV130 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[461]) ).
thf(586,plain,
! [SV131: $i,SV8: $i] :
( ( ! [SY393: $i] :
( ~ ( member @ SV8 @ ( unordered_pair @ SV131 @ SY393 ) )
| ( SV8 = SV131 )
| ( SV8 = SY393 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[462]) ).
thf(587,plain,
! [SV9: $i] :
( ( ( ~ ( member @ SV9 @ universal_class ) )
= $true )
| ( ( ! [SY267: $i] : ( member @ SV9 @ ( unordered_pair @ SV9 @ SY267 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[463]) ).
thf(588,plain,
! [SV10: $i] :
( ( ( ~ ( member @ SV10 @ universal_class ) )
= $true )
| ( ( ! [SY268: $i] : ( member @ SV10 @ ( unordered_pair @ SY268 @ SV10 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[464]) ).
thf(589,plain,
! [SV132: $i,SV11: $i] :
( ( member @ ( unordered_pair @ SV11 @ SV132 ) @ universal_class )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[465]) ).
thf(590,plain,
! [SV133: $i,SV13: $i] :
( ( ( unordered_pair @ ( singleton @ SV13 ) @ ( unordered_pair @ SV13 @ ( singleton @ SV133 ) ) )
= ( ordered_pair @ SV13 @ SV133 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[467]) ).
thf(591,plain,
! [SV134: $i,SV14: $i] :
( ( ! [SY394: $i] :
( ! [SY395: $i] :
~ ( member @ ( ordered_pair @ SV14 @ SV134 ) @ ( cross_product @ SY394 @ SY395 ) )
| ( member @ SV14 @ SY394 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[468]) ).
thf(592,plain,
! [SV135: $i,SV15: $i] :
( ( ! [SY396: $i,SY397: $i] :
( ~ ( member @ ( ordered_pair @ SV15 @ SV135 ) @ ( cross_product @ SY396 @ SY397 ) )
| ( member @ SV135 @ SY397 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[469]) ).
thf(593,plain,
! [SV136: $i,SV16: $i] :
( ( ~ ( member @ SV16 @ SV136 )
| ! [SY398: $i,SY399: $i] :
( ~ ( member @ SY398 @ SY399 )
| ( member @ ( ordered_pair @ SV16 @ SY398 ) @ ( cross_product @ SV136 @ SY399 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[470]) ).
thf(594,plain,
! [SV17: $i] :
( ( ( ! [SY280: $i,SY281: $i] :
~ ( member @ SV17 @ ( cross_product @ SY280 @ SY281 ) ) )
= $true )
| ( ( ( ordered_pair @ ( first @ SV17 ) @ ( second @ SV17 ) )
= SV17 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[471]) ).
thf(595,plain,
! [SV137: $i,SV18: $i] :
( ( ~ ( member @ ( ordered_pair @ SV18 @ SV137 ) @ element_relation )
| ( member @ SV18 @ SV137 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[472]) ).
thf(596,plain,
! [SV138: $i,SV19: $i] :
( ( ~ ( member @ ( ordered_pair @ SV19 @ SV138 ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ SV19 @ SV138 )
| ( member @ ( ordered_pair @ SV19 @ SV138 ) @ element_relation ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[473]) ).
thf(597,plain,
! [SV139: $i,SV20: $i] :
( ( ! [SY400: $i] :
~ ( member @ SV20 @ ( intersection @ SV139 @ SY400 ) )
| ( member @ SV20 @ SV139 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[474]) ).
thf(598,plain,
! [SV140: $i,SV21: $i] :
( ( ! [SY401: $i] :
( ~ ( member @ SV21 @ ( intersection @ SV140 @ SY401 ) )
| ( member @ SV21 @ SY401 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[475]) ).
thf(599,plain,
! [SV141: $i,SV22: $i] :
( ( ~ ( member @ SV22 @ SV141 )
| ! [SY402: $i] :
( ~ ( member @ SV22 @ SY402 )
| ( member @ SV22 @ ( intersection @ SV141 @ SY402 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[476]) ).
thf(600,plain,
! [SV142: $i,SV23: $i] :
( ( ~ ( member @ SV23 @ ( complement @ SV142 ) )
| ~ ( member @ SV23 @ SV142 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[477]) ).
thf(601,plain,
! [SV24: $i] :
( ( ( ~ ( member @ SV24 @ universal_class ) )
= $true )
| ( ( ! [SY291: $i] :
( ( member @ SV24 @ ( complement @ SY291 ) )
| ( member @ SV24 @ SY291 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[478]) ).
thf(602,plain,
! [SV143: $i,SV25: $i] :
( ( ( complement @ ( intersection @ ( complement @ SV25 ) @ ( complement @ SV143 ) ) )
= ( union @ SV25 @ SV143 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[479]) ).
thf(603,plain,
! [SV144: $i,SV26: $i] :
( ( ( intersection @ ( complement @ ( intersection @ SV26 @ SV144 ) ) @ ( complement @ ( intersection @ ( complement @ SV26 ) @ ( complement @ SV144 ) ) ) )
= ( symmetric_difference @ SV26 @ SV144 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[480]) ).
thf(604,plain,
! [SV145: $i,SV27: $i] :
( ( ! [SY403: $i] :
( ( intersection @ SV27 @ ( cross_product @ SV145 @ SY403 ) )
= ( restrict @ SV27 @ SV145 @ SY403 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[481]) ).
thf(605,plain,
! [SV146: $i,SV28: $i] :
( ( ! [SY404: $i] :
( ( intersection @ ( cross_product @ SV28 @ SV146 ) @ SY404 )
= ( restrict @ SY404 @ SV28 @ SV146 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[482]) ).
thf(606,plain,
! [SV147: $i,SV29: $i] :
( ( ( ( restrict @ SV29 @ ( singleton @ SV147 ) @ universal_class )
!= null_class )
| ~ ( member @ SV147 @ ( domain_of @ SV29 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[483]) ).
thf(607,plain,
! [SV30: $i] :
( ( ( ~ ( member @ SV30 @ universal_class ) )
= $true )
| ( ( ! [SY299: $i] :
( ( ( restrict @ SY299 @ ( singleton @ SV30 ) @ universal_class )
= null_class )
| ( member @ SV30 @ ( domain_of @ SY299 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[484]) ).
thf(608,plain,
! [SV148: $i,SV32: $i] :
( ( ! [SY405: $i,SY406: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV148 ) @ SY405 ) @ ( rotate @ SY406 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV148 @ SY405 ) @ SV32 ) @ SY406 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[486]) ).
thf(609,plain,
! [SV149: $i,SV33: $i] :
( ( ! [SY407: $i,SY408: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV149 ) @ SY407 ) @ SY408 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SY407 @ SV33 ) @ SV149 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY407 @ SV33 ) @ SV149 ) @ ( rotate @ SY408 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[487]) ).
thf(610,plain,
! [SV150: $i,SV35: $i] :
( ( ! [SY409: $i,SY410: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV35 @ SV150 ) @ SY409 ) @ ( flip @ SY410 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV150 @ SV35 ) @ SY409 ) @ SY410 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[489]) ).
thf(611,plain,
! [SV151: $i,SV36: $i] :
( ( ! [SY411: $i,SY412: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV36 @ SV151 ) @ SY411 ) @ SY412 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SY411 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SY411 ) @ ( flip @ SY412 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[490]) ).
thf(612,plain,
! [SV152: $i,SV39: $i] :
( ( ! [SY413: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ SV39 @ SV152 @ ( singleton @ SY413 ) ) @ null_class ) )
= ( domain @ SV39 @ SV152 @ SY413 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[493]) ).
thf(613,plain,
! [SV153: $i,SV40: $i] :
( ( ! [SY414: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ SV40 @ ( singleton @ SV153 ) @ SY414 ) @ null_class ) )
= ( range @ SV40 @ SV153 @ SY414 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[494]) ).
thf(614,plain,
! [SV154: $i,SV41: $i] :
( ( ( range_of @ ( restrict @ SV41 @ SV154 @ universal_class ) )
= ( image @ SV41 @ SV154 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[495]) ).
thf(615,plain,
! [SV155: $i,SV43: $i] :
( ( ~ ( member @ ( ordered_pair @ SV43 @ SV155 ) @ successor_relation )
| ( ( successor @ SV43 )
= SV155 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[497]) ).
thf(616,plain,
! [SV156: $i,SV44: $i] :
( ( ( ( successor @ SV44 )
!= SV156 )
| ~ ( member @ ( ordered_pair @ SV44 @ SV156 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV44 @ SV156 ) @ successor_relation ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[498]) ).
thf(617,plain,
! [SV45: $i] :
( ( ( ~ ( inductive @ SV45 ) )
= $true )
| ( ( member @ null_class @ SV45 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[499]) ).
thf(618,plain,
! [SV46: $i] :
( ( ( ~ ( inductive @ SV46 ) )
= $true )
| ( ( subclass @ ( image @ successor_relation @ SV46 ) @ SV46 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[500]) ).
thf(619,plain,
! [SV47: $i] :
( ( ( ~ ( member @ null_class @ SV47 ) )
= $true )
| ( ( ~ ( subclass @ ( image @ successor_relation @ SV47 ) @ SV47 )
| ( inductive @ SV47 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[501]) ).
thf(620,plain,
! [SV48: $i] :
( ( ( ~ ( inductive @ SV48 ) )
= $true )
| ( ( subclass @ omega @ SV48 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[502]) ).
thf(621,plain,
! [SV50: $i] :
( ( ( ~ ( member @ SV50 @ universal_class ) )
= $true )
| ( ( member @ ( sum_class @ SV50 ) @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[504]) ).
thf(622,plain,
! [SV52: $i] :
( ( ( ~ ( member @ SV52 @ universal_class ) )
= $true )
| ( ( member @ ( power_class @ SV52 ) @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[506]) ).
thf(623,plain,
! [SV157: $i,SV53: $i] :
( ( subclass @ ( compose @ SV53 @ SV157 ) @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[507]) ).
thf(624,plain,
! [SV158: $i,SV54: $i] :
( ( ! [SY415: $i,SY416: $i] :
( ~ ( member @ ( ordered_pair @ SV54 @ SV158 ) @ ( compose @ SY415 @ SY416 ) )
| ( member @ SV158 @ ( image @ SY415 @ ( image @ SY416 @ ( singleton @ SV54 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[508]) ).
thf(625,plain,
! [SV159: $i,SV55: $i] :
( ( ! [SY417: $i,SY418: $i] :
( ~ ( member @ SV55 @ ( image @ SV159 @ ( image @ SY417 @ ( singleton @ SY418 ) ) ) )
| ~ ( member @ ( ordered_pair @ SY418 @ SV55 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SY418 @ SV55 ) @ ( compose @ SV159 @ SY417 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[509]) ).
thf(626,plain,
! [SV56: $i] :
( ( ( ~ ( single_valued_class @ SV56 ) )
= $true )
| ( ( subclass @ ( compose @ SV56 @ ( inverse @ SV56 ) ) @ identity_relation )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[510]) ).
thf(627,plain,
! [SV57: $i] :
( ( ( ~ ( subclass @ ( compose @ SV57 @ ( inverse @ SV57 ) ) @ identity_relation ) )
= $true )
| ( ( single_valued_class @ SV57 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[511]) ).
thf(628,plain,
! [SV58: $i] :
( ( ( ~ ( function @ SV58 ) )
= $true )
| ( ( subclass @ SV58 @ ( cross_product @ universal_class @ universal_class ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[512]) ).
thf(629,plain,
! [SV59: $i] :
( ( ( ~ ( function @ SV59 ) )
= $true )
| ( ( subclass @ ( compose @ SV59 @ ( inverse @ SV59 ) ) @ identity_relation )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[513]) ).
thf(630,plain,
! [SV60: $i] :
( ( ( ~ ( subclass @ SV60 @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( ~ ( subclass @ ( compose @ SV60 @ ( inverse @ SV60 ) ) @ identity_relation )
| ( function @ SV60 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[514]) ).
thf(631,plain,
! [SV61: $i] :
( ( ( ~ ( function @ SV61 ) )
= $true )
| ( ( ! [SY326: $i] :
( ~ ( member @ SY326 @ universal_class )
| ( member @ ( image @ SV61 @ SY326 ) @ universal_class ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[515]) ).
thf(632,plain,
! [SV62: $i] :
( ( ( SV62 = null_class )
= $true )
| ( ( member @ ( regular @ SV62 ) @ SV62 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[516]) ).
thf(633,plain,
! [SV63: $i] :
( ( ( SV63 = null_class )
= $true )
| ( ( ( intersection @ SV63 @ ( regular @ SV63 ) )
= null_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[517]) ).
thf(634,plain,
! [SV160: $i,SV64: $i] :
( ( ( sum_class @ ( image @ SV64 @ ( singleton @ SV160 ) ) )
= ( apply @ SV64 @ SV160 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[518]) ).
thf(635,plain,
! [SV65: $i] :
( ( ( ~ ( member @ SV65 @ universal_class ) )
= $true )
| ( ( ( SV65 = null_class )
| ( member @ ( apply @ choice @ SV65 ) @ SV65 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[519]) ).
thf(636,plain,
! [SV66: $i] :
( ( ( ~ ( one_to_one @ SV66 ) )
= $true )
| ( ( function @ SV66 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[520]) ).
thf(637,plain,
! [SV67: $i] :
( ( ( ~ ( one_to_one @ SV67 ) )
= $true )
| ( ( function @ ( inverse @ SV67 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[521]) ).
thf(638,plain,
! [SV68: $i] :
( ( ( ~ ( function @ ( inverse @ SV68 ) ) )
= $true )
| ( ( ~ ( function @ SV68 )
| ( one_to_one @ SV68 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[522]) ).
thf(639,plain,
! [SV71: $i] :
( ( ( ~ ( operation @ SV71 ) )
= $true )
| ( ( function @ SV71 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[525]) ).
thf(640,plain,
! [SV72: $i] :
( ( ( ~ ( operation @ SV72 ) )
= $true )
| ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV72 ) ) @ ( domain_of @ ( domain_of @ SV72 ) ) )
= ( domain_of @ SV72 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[526]) ).
thf(641,plain,
! [SV73: $i] :
( ( ( ~ ( operation @ SV73 ) )
= $true )
| ( ( subclass @ ( range_of @ SV73 ) @ ( domain_of @ ( domain_of @ SV73 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[527]) ).
thf(642,plain,
! [SV74: $i] :
( ( ( ~ ( function @ SV74 ) )
= $true )
| ( ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV74 ) ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
!= ( domain_of @ SV74 ) )
| ~ ( subclass @ ( range_of @ SV74 ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
| ( operation @ SV74 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[528]) ).
thf(643,plain,
! [SV75: $i] :
( ( ( ! [SY328: $i,SY329: $i] :
~ ( compatible @ SV75 @ SY328 @ SY329 ) )
= $true )
| ( ( function @ SV75 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[529]) ).
thf(644,plain,
! [SV161: $i,SV76: $i] :
( ( ! [SY419: $i] :
~ ( compatible @ SV76 @ SV161 @ SY419 )
| ( ( domain_of @ ( domain_of @ SV161 ) )
= ( domain_of @ SV76 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[530]) ).
thf(645,plain,
! [SV162: $i,SV77: $i] :
( ( ! [SY420: $i] :
( ~ ( compatible @ SV77 @ SV162 @ SY420 )
| ( subclass @ ( range_of @ SV77 ) @ ( domain_of @ ( domain_of @ SY420 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[531]) ).
thf(646,plain,
! [SV78: $i] :
( ( ( ~ ( function @ SV78 ) )
= $true )
| ( ( ! [SY334: $i] :
( ( ( domain_of @ ( domain_of @ SY334 ) )
!= ( domain_of @ SV78 ) )
| ! [SY335: $i] :
( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SY335 ) ) )
| ( compatible @ SV78 @ SY334 @ SY335 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[532]) ).
thf(647,plain,
! [SV163: $i,SV79: $i] :
( ( ! [SY421: $i] :
~ ( homomorphism @ SV79 @ SV163 @ SY421 )
| ( operation @ SV163 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[533]) ).
thf(648,plain,
! [SV164: $i,SV80: $i] :
( ( ! [SY422: $i] :
( ~ ( homomorphism @ SV80 @ SV164 @ SY422 )
| ( operation @ SY422 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[534]) ).
thf(649,plain,
! [SV165: $i,SV81: $i] :
( ( ! [SY423: $i] :
( ~ ( homomorphism @ SV81 @ SV165 @ SY423 )
| ( compatible @ SV81 @ SV165 @ SY423 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[535]) ).
thf(650,plain,
! [SV166: $i,SV82: $i] :
( ( ! [SY424: $i] :
( ~ ( homomorphism @ SV82 @ SV166 @ SY424 )
| ! [SY425: $i,SY426: $i] :
( ~ ( member @ ( ordered_pair @ SY425 @ SY426 ) @ ( domain_of @ SV166 ) )
| ( ( apply @ SY424 @ ( ordered_pair @ ( apply @ SV82 @ SY425 ) @ ( apply @ SV82 @ SY426 ) ) )
= ( apply @ SV82 @ ( apply @ SV166 @ ( ordered_pair @ SY425 @ SY426 ) ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[536]) ).
thf(651,plain,
! [SV83: $i] :
( ( ( ~ ( operation @ SV83 ) )
= $true )
| ( ( ! [SY346: $i] :
( ~ ( operation @ SY346 )
| ! [SY347: $i] :
( ~ ( compatible @ SY347 @ SV83 @ SY346 )
| ( homomorphism @ SY347 @ SV83 @ SY346 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SY347 @ SV83 @ SY346 ) @ ( not_homomorphism2 @ SY347 @ SV83 @ SY346 ) ) @ ( domain_of @ SV83 ) ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[537]) ).
thf(652,plain,
! [SV84: $i] :
( ( ( ~ ( operation @ SV84 ) )
= $true )
| ( ( ! [SY348: $i] :
( ~ ( operation @ SY348 )
| ! [SY349: $i] :
( ~ ( compatible @ SY349 @ SV84 @ SY348 )
| ( ( apply @ SY348 @ ( ordered_pair @ ( apply @ SY349 @ ( not_homomorphism1 @ SY349 @ SV84 @ SY348 ) ) @ ( apply @ SY349 @ ( not_homomorphism2 @ SY349 @ SV84 @ SY348 ) ) ) )
!= ( apply @ SY349 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SY349 @ SV84 @ SY348 ) @ ( not_homomorphism2 @ SY349 @ SV84 @ SY348 ) ) ) ) )
| ( homomorphism @ SY349 @ SV84 @ SY348 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[538]) ).
thf(653,plain,
! [SV167: $i,SV85: $i] :
( ( ! [SY427: $i,SY428: $i] :
~ ( member @ ( ordered_pair @ SV85 @ SV167 ) @ ( cross_product @ SY427 @ SY428 ) )
| ( member @ SV85 @ ( unordered_pair @ SV85 @ SV167 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[539]) ).
thf(654,plain,
! [SV168: $i,SV86: $i] :
( ( ! [SY429: $i,SY430: $i] :
~ ( member @ ( ordered_pair @ SV86 @ SV168 ) @ ( cross_product @ SY429 @ SY430 ) )
| ( member @ SV168 @ ( unordered_pair @ SV86 @ SV168 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[540]) ).
thf(655,plain,
! [SV169: $i,SV87: $i] :
( ( ! [SY431: $i,SY432: $i] :
~ ( member @ ( ordered_pair @ SV87 @ SV169 ) @ ( cross_product @ SY431 @ SY432 ) )
| ( member @ SV87 @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[541]) ).
thf(656,plain,
! [SV170: $i,SV88: $i] :
( ( ! [SY433: $i,SY434: $i] :
~ ( member @ ( ordered_pair @ SV88 @ SV170 ) @ ( cross_product @ SY433 @ SY434 ) )
| ( member @ SV170 @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[542]) ).
thf(657,plain,
! [SV171: $i,SV90: $i] :
( ( ~ ( subclass @ SV90 @ SV171 )
| ! [SY435: $i] :
( ~ ( subclass @ SV171 @ SY435 )
| ( subclass @ SV90 @ SY435 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[544]) ).
thf(658,plain,
! [SV172: $i,SV91: $i] :
( ( ( SV91 = SV172 )
| ( member @ ( not_subclass_element @ SV91 @ SV172 ) @ SV91 )
| ( member @ ( not_subclass_element @ SV172 @ SV91 ) @ SV172 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[545]) ).
thf(659,plain,
! [SV173: $i,SV92: $i] :
( ( ~ ( member @ ( not_subclass_element @ SV92 @ SV173 ) @ SV173 )
| ( SV92 = SV173 )
| ( member @ ( not_subclass_element @ SV173 @ SV92 ) @ SV173 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[546]) ).
thf(660,plain,
! [SV174: $i,SV93: $i] :
( ( ~ ( member @ ( not_subclass_element @ SV93 @ SV174 ) @ SV174 )
| ( SV174 = SV93 )
| ( member @ ( not_subclass_element @ SV174 @ SV93 ) @ SV174 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[547]) ).
thf(661,plain,
! [SV175: $i,SV94: $i] :
( ( ~ ( member @ ( not_subclass_element @ SV94 @ SV175 ) @ SV175 )
| ~ ( member @ ( not_subclass_element @ SV175 @ SV94 ) @ SV94 )
| ( SV94 = SV175 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[548]) ).
thf(662,plain,
! [SV176: $i,SV95: $i] :
( ( ~ ( member @ SV95 @ ( intersection @ ( complement @ SV176 ) @ SV176 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[549]) ).
thf(663,plain,
! [SV96: $i] :
( ( member @ SV96 @ null_class )
= $false ),
inference(extcnf_not_pos,[status(thm)],[550]) ).
thf(664,plain,
! [SV98: $i] :
( ( ( ~ ( subclass @ SV98 @ null_class ) )
= $true )
| ( ( SV98 = null_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[552]) ).
thf(665,plain,
! [SV99: $i] :
( ( ( SV99 = null_class )
= $true )
| ( ( member @ ( not_subclass_element @ SV99 @ null_class ) @ SV99 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[553]) ).
thf(666,plain,
! [SV177: $i,SV100: $i] :
( ( ( unordered_pair @ SV100 @ SV177 )
= ( unordered_pair @ SV177 @ SV100 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[554]) ).
thf(667,plain,
! [SV178: $i,SV101: $i] :
( ( subclass @ ( singleton @ SV101 ) @ ( unordered_pair @ SV101 @ SV178 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[555]) ).
thf(668,plain,
! [SV179: $i,SV102: $i] :
( ( subclass @ ( singleton @ SV102 ) @ ( unordered_pair @ SV179 @ SV102 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[556]) ).
thf(669,plain,
! [SV103: $i] :
( ( ( ! [SY372: $i] :
( ( unordered_pair @ SY372 @ SV103 )
= ( singleton @ SY372 ) ) )
= $true )
| ( ( member @ SV103 @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[557]) ).
thf(670,plain,
! [SV104: $i] :
( ( ( ! [SY373: $i] :
( ( unordered_pair @ SV104 @ SY373 )
= ( singleton @ SY373 ) ) )
= $true )
| ( ( member @ SV104 @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[558]) ).
thf(671,plain,
! [SV180: $i,SV105: $i] :
( ( ( ( unordered_pair @ SV105 @ SV180 )
= null_class )
| ( member @ SV105 @ universal_class )
| ( member @ SV180 @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[559]) ).
thf(672,plain,
! [SV181: $i,SV106: $i] :
( ( ! [SY436: $i] :
( ( ( unordered_pair @ SV106 @ SV181 )
!= ( unordered_pair @ SV106 @ SY436 ) )
| ~ ( member @ ( ordered_pair @ SV181 @ SY436 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV181 = SY436 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[560]) ).
thf(673,plain,
! [SV182: $i,SV107: $i] :
( ( ! [SY437: $i] :
( ( ( unordered_pair @ SV107 @ SV182 )
!= ( unordered_pair @ SY437 @ SV182 ) )
| ~ ( member @ ( ordered_pair @ SV107 @ SY437 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV107 = SY437 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[561]) ).
thf(674,plain,
! [SV108: $i] :
( ( ( ~ ( member @ SV108 @ universal_class ) )
= $true )
| ( ( ! [SY379: $i] :
( ( unordered_pair @ SV108 @ SY379 )
!= null_class ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[562]) ).
thf(675,plain,
! [SV109: $i] :
( ( ( ~ ( member @ SV109 @ universal_class ) )
= $true )
| ( ( ! [SY380: $i] :
( ( unordered_pair @ SY380 @ SV109 )
!= null_class ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[563]) ).
thf(676,plain,
! [SV183: $i,SV110: $i] :
( ( ! [SY438: $i,SY439: $i] :
~ ( member @ ( ordered_pair @ SV110 @ SV183 ) @ ( cross_product @ SY438 @ SY439 ) )
| ( ( unordered_pair @ SV110 @ SV183 )
!= null_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[564]) ).
thf(677,plain,
! [SV184: $i,SV111: $i] :
( ( ~ ( member @ SV111 @ SV184 )
| ! [SY440: $i] :
( ~ ( member @ SY440 @ SV184 )
| ( subclass @ ( unordered_pair @ SV111 @ SY440 ) @ SV184 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[565]) ).
thf(678,plain,
! [SV185: $i,SV113: $i] :
( ( member @ ( singleton @ SV113 ) @ ( unordered_pair @ SV185 @ ( singleton @ SV113 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[567]) ).
thf(679,plain,
! [SV114: $i] :
( ( ( ~ ( member @ SV114 @ universal_class ) )
= $true )
| ( ( member @ SV114 @ ( singleton @ SV114 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[568]) ).
thf(680,plain,
! [SV115: $i] :
( ( ( ~ ( member @ SV115 @ universal_class ) )
= $true )
| ( ( ( singleton @ SV115 )
!= null_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[569]) ).
thf(681,plain,
! [SV186: $i,SV116: $i] :
( ( ~ ( member @ SV116 @ ( singleton @ SV186 ) )
| ( SV116 = SV186 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[570]) ).
thf(682,plain,
! [SV117: $i] :
( ( ( ( singleton @ SV117 )
= null_class )
= $true )
| ( ( member @ SV117 @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[571]) ).
thf(683,plain,
! [SV187: $i,SV118: $i] :
( ( ( ( singleton @ SV118 )
!= ( singleton @ SV187 ) )
| ~ ( member @ SV118 @ universal_class )
| ( SV118 = SV187 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[572]) ).
thf(684,plain,
! [SV188: $i,SV119: $i] :
( ( ( ( singleton @ SV119 )
!= ( singleton @ SV188 ) )
| ~ ( member @ SV188 @ universal_class )
| ( SV119 = SV188 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[573]) ).
thf(685,plain,
! [SV189: $i,SV120: $i] :
( ( ! [SY441: $i] :
( ( ( unordered_pair @ SV120 @ SV189 )
!= ( singleton @ SY441 ) )
| ~ ( member @ SY441 @ universal_class )
| ( SY441 = SV120 )
| ( SY441 = SV189 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[574]) ).
thf(686,plain,
! [SV121: $i] :
( ( ( ~ ( member @ SV121 @ universal_class ) )
= $true )
| ( ( member @ ( member_of @ ( singleton @ SV121 ) ) @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[575]) ).
thf(687,plain,
! [SV122: $i] :
( ( ( ~ ( member @ SV122 @ universal_class ) )
= $true )
| ( ( ( singleton @ ( member_of @ ( singleton @ SV122 ) ) )
= ( singleton @ SV122 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[576]) ).
thf(688,plain,
! [SV123: $i] :
( ( ( ( member_of @ SV123 )
= SV123 )
= $true )
| ( ( member @ ( member_of @ SV123 ) @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[577]) ).
thf(689,plain,
! [SV124: $i] :
( ( ( ( singleton @ ( member_of @ SV124 ) )
= SV124 )
= $true )
| ( ( ( member_of @ SV124 )
= SV124 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[578]) ).
thf(690,plain,
! [SV125: $i,SV1: $i] :
( ( ( ~ ( subclass @ SV1 @ SV125 ) )
= $true )
| ( ( ! [SY392: $i] :
( ~ ( member @ SY392 @ SV1 )
| ( member @ SY392 @ SV125 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[580]) ).
thf(691,plain,
! [SV126: $i,SV2: $i] :
( ( ( member @ ( not_subclass_element @ SV2 @ SV126 ) @ SV2 )
= $true )
| ( ( subclass @ SV2 @ SV126 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[581]) ).
thf(692,plain,
! [SV127: $i,SV3: $i] :
( ( ( ~ ( member @ ( not_subclass_element @ SV3 @ SV127 ) @ SV127 ) )
= $true )
| ( ( subclass @ SV3 @ SV127 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[582]) ).
thf(693,plain,
! [SV128: $i,SV5: $i] :
( ( ( SV5 != SV128 )
= $true )
| ( ( subclass @ SV5 @ SV128 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[583]) ).
thf(694,plain,
! [SV129: $i,SV6: $i] :
( ( ( SV6 != SV129 )
= $true )
| ( ( subclass @ SV129 @ SV6 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[584]) ).
thf(695,plain,
! [SV130: $i,SV7: $i] :
( ( ( ~ ( subclass @ SV7 @ SV130 ) )
= $true )
| ( ( ~ ( subclass @ SV130 @ SV7 )
| ( SV7 = SV130 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[585]) ).
thf(696,plain,
! [SV190: $i,SV131: $i,SV8: $i] :
( ( ~ ( member @ SV8 @ ( unordered_pair @ SV131 @ SV190 ) )
| ( SV8 = SV131 )
| ( SV8 = SV190 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[586]) ).
thf(697,plain,
! [SV9: $i] :
( ( ( member @ SV9 @ universal_class )
= $false )
| ( ( ! [SY267: $i] : ( member @ SV9 @ ( unordered_pair @ SV9 @ SY267 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[587]) ).
thf(698,plain,
! [SV10: $i] :
( ( ( member @ SV10 @ universal_class )
= $false )
| ( ( ! [SY268: $i] : ( member @ SV10 @ ( unordered_pair @ SY268 @ SV10 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[588]) ).
thf(699,plain,
! [SV191: $i,SV134: $i,SV14: $i] :
( ( ! [SY442: $i] :
~ ( member @ ( ordered_pair @ SV14 @ SV134 ) @ ( cross_product @ SV191 @ SY442 ) )
| ( member @ SV14 @ SV191 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[591]) ).
thf(700,plain,
! [SV192: $i,SV135: $i,SV15: $i] :
( ( ! [SY443: $i] :
( ~ ( member @ ( ordered_pair @ SV15 @ SV135 ) @ ( cross_product @ SV192 @ SY443 ) )
| ( member @ SV135 @ SY443 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[592]) ).
thf(701,plain,
! [SV136: $i,SV16: $i] :
( ( ( ~ ( member @ SV16 @ SV136 ) )
= $true )
| ( ( ! [SY398: $i,SY399: $i] :
( ~ ( member @ SY398 @ SY399 )
| ( member @ ( ordered_pair @ SV16 @ SY398 ) @ ( cross_product @ SV136 @ SY399 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[593]) ).
thf(702,plain,
! [SV193: $i,SV17: $i] :
( ( ( ! [SY444: $i] :
~ ( member @ SV17 @ ( cross_product @ SV193 @ SY444 ) ) )
= $true )
| ( ( ( ordered_pair @ ( first @ SV17 ) @ ( second @ SV17 ) )
= SV17 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[594]) ).
thf(703,plain,
! [SV137: $i,SV18: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV18 @ SV137 ) @ element_relation ) )
= $true )
| ( ( member @ SV18 @ SV137 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[595]) ).
thf(704,plain,
! [SV138: $i,SV19: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV19 @ SV138 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( ~ ( member @ SV19 @ SV138 )
| ( member @ ( ordered_pair @ SV19 @ SV138 ) @ element_relation ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[596]) ).
thf(705,plain,
! [SV139: $i,SV20: $i] :
( ( ( ! [SY400: $i] :
~ ( member @ SV20 @ ( intersection @ SV139 @ SY400 ) ) )
= $true )
| ( ( member @ SV20 @ SV139 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[597]) ).
thf(706,plain,
! [SV194: $i,SV140: $i,SV21: $i] :
( ( ~ ( member @ SV21 @ ( intersection @ SV140 @ SV194 ) )
| ( member @ SV21 @ SV194 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[598]) ).
thf(707,plain,
! [SV141: $i,SV22: $i] :
( ( ( ~ ( member @ SV22 @ SV141 ) )
= $true )
| ( ( ! [SY402: $i] :
( ~ ( member @ SV22 @ SY402 )
| ( member @ SV22 @ ( intersection @ SV141 @ SY402 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[599]) ).
thf(708,plain,
! [SV142: $i,SV23: $i] :
( ( ( ~ ( member @ SV23 @ ( complement @ SV142 ) ) )
= $true )
| ( ( ~ ( member @ SV23 @ SV142 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[600]) ).
thf(709,plain,
! [SV24: $i] :
( ( ( member @ SV24 @ universal_class )
= $false )
| ( ( ! [SY291: $i] :
( ( member @ SV24 @ ( complement @ SY291 ) )
| ( member @ SV24 @ SY291 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[601]) ).
thf(710,plain,
! [SV195: $i,SV145: $i,SV27: $i] :
( ( ( intersection @ SV27 @ ( cross_product @ SV145 @ SV195 ) )
= ( restrict @ SV27 @ SV145 @ SV195 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[604]) ).
thf(711,plain,
! [SV196: $i,SV146: $i,SV28: $i] :
( ( ( intersection @ ( cross_product @ SV28 @ SV146 ) @ SV196 )
= ( restrict @ SV196 @ SV28 @ SV146 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[605]) ).
thf(712,plain,
! [SV147: $i,SV29: $i] :
( ( ( ( restrict @ SV29 @ ( singleton @ SV147 ) @ universal_class )
!= null_class )
= $true )
| ( ( ~ ( member @ SV147 @ ( domain_of @ SV29 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[606]) ).
thf(713,plain,
! [SV30: $i] :
( ( ( member @ SV30 @ universal_class )
= $false )
| ( ( ! [SY299: $i] :
( ( ( restrict @ SY299 @ ( singleton @ SV30 ) @ universal_class )
= null_class )
| ( member @ SV30 @ ( domain_of @ SY299 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[607]) ).
thf(714,plain,
! [SV197: $i,SV148: $i,SV32: $i] :
( ( ! [SY445: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV148 ) @ SV197 ) @ ( rotate @ SY445 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV148 @ SV197 ) @ SV32 ) @ SY445 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[608]) ).
thf(715,plain,
! [SV198: $i,SV149: $i,SV33: $i] :
( ( ! [SY446: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV149 ) @ SV198 ) @ SY446 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( rotate @ SY446 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[609]) ).
thf(716,plain,
! [SV199: $i,SV150: $i,SV35: $i] :
( ( ! [SY447: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV35 @ SV150 ) @ SV199 ) @ ( flip @ SY447 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV150 @ SV35 ) @ SV199 ) @ SY447 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[610]) ).
thf(717,plain,
! [SV200: $i,SV151: $i,SV36: $i] :
( ( ! [SY448: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV36 @ SV151 ) @ SV200 ) @ SY448 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( flip @ SY448 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[611]) ).
thf(718,plain,
! [SV201: $i,SV152: $i,SV39: $i] :
( ( ( first @ ( not_subclass_element @ ( restrict @ SV39 @ SV152 @ ( singleton @ SV201 ) ) @ null_class ) )
= ( domain @ SV39 @ SV152 @ SV201 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[612]) ).
thf(719,plain,
! [SV202: $i,SV153: $i,SV40: $i] :
( ( ( second @ ( not_subclass_element @ ( restrict @ SV40 @ ( singleton @ SV153 ) @ SV202 ) @ null_class ) )
= ( range @ SV40 @ SV153 @ SV202 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[613]) ).
thf(720,plain,
! [SV155: $i,SV43: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV43 @ SV155 ) @ successor_relation ) )
= $true )
| ( ( ( successor @ SV43 )
= SV155 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[615]) ).
thf(721,plain,
! [SV156: $i,SV44: $i] :
( ( ( ( successor @ SV44 )
!= SV156 )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ SV44 @ SV156 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV44 @ SV156 ) @ successor_relation ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[616]) ).
thf(722,plain,
! [SV45: $i] :
( ( ( inductive @ SV45 )
= $false )
| ( ( member @ null_class @ SV45 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[617]) ).
thf(723,plain,
! [SV46: $i] :
( ( ( inductive @ SV46 )
= $false )
| ( ( subclass @ ( image @ successor_relation @ SV46 ) @ SV46 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[618]) ).
thf(724,plain,
! [SV47: $i] :
( ( ( member @ null_class @ SV47 )
= $false )
| ( ( ~ ( subclass @ ( image @ successor_relation @ SV47 ) @ SV47 )
| ( inductive @ SV47 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[619]) ).
thf(725,plain,
! [SV48: $i] :
( ( ( inductive @ SV48 )
= $false )
| ( ( subclass @ omega @ SV48 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[620]) ).
thf(726,plain,
! [SV50: $i] :
( ( ( member @ SV50 @ universal_class )
= $false )
| ( ( member @ ( sum_class @ SV50 ) @ universal_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[621]) ).
thf(727,plain,
! [SV52: $i] :
( ( ( member @ SV52 @ universal_class )
= $false )
| ( ( member @ ( power_class @ SV52 ) @ universal_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[622]) ).
thf(728,plain,
! [SV203: $i,SV158: $i,SV54: $i] :
( ( ! [SY449: $i] :
( ~ ( member @ ( ordered_pair @ SV54 @ SV158 ) @ ( compose @ SV203 @ SY449 ) )
| ( member @ SV158 @ ( image @ SV203 @ ( image @ SY449 @ ( singleton @ SV54 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[624]) ).
thf(729,plain,
! [SV204: $i,SV159: $i,SV55: $i] :
( ( ! [SY450: $i] :
( ~ ( member @ SV55 @ ( image @ SV159 @ ( image @ SV204 @ ( singleton @ SY450 ) ) ) )
| ~ ( member @ ( ordered_pair @ SY450 @ SV55 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SY450 @ SV55 ) @ ( compose @ SV159 @ SV204 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[625]) ).
thf(730,plain,
! [SV56: $i] :
( ( ( single_valued_class @ SV56 )
= $false )
| ( ( subclass @ ( compose @ SV56 @ ( inverse @ SV56 ) ) @ identity_relation )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[626]) ).
thf(731,plain,
! [SV57: $i] :
( ( ( subclass @ ( compose @ SV57 @ ( inverse @ SV57 ) ) @ identity_relation )
= $false )
| ( ( single_valued_class @ SV57 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[627]) ).
thf(732,plain,
! [SV58: $i] :
( ( ( function @ SV58 )
= $false )
| ( ( subclass @ SV58 @ ( cross_product @ universal_class @ universal_class ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[628]) ).
thf(733,plain,
! [SV59: $i] :
( ( ( function @ SV59 )
= $false )
| ( ( subclass @ ( compose @ SV59 @ ( inverse @ SV59 ) ) @ identity_relation )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[629]) ).
thf(734,plain,
! [SV60: $i] :
( ( ( subclass @ SV60 @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( ~ ( subclass @ ( compose @ SV60 @ ( inverse @ SV60 ) ) @ identity_relation )
| ( function @ SV60 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[630]) ).
thf(735,plain,
! [SV61: $i] :
( ( ( function @ SV61 )
= $false )
| ( ( ! [SY326: $i] :
( ~ ( member @ SY326 @ universal_class )
| ( member @ ( image @ SV61 @ SY326 ) @ universal_class ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[631]) ).
thf(736,plain,
! [SV65: $i] :
( ( ( member @ SV65 @ universal_class )
= $false )
| ( ( ( SV65 = null_class )
| ( member @ ( apply @ choice @ SV65 ) @ SV65 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[635]) ).
thf(737,plain,
! [SV66: $i] :
( ( ( one_to_one @ SV66 )
= $false )
| ( ( function @ SV66 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[636]) ).
thf(738,plain,
! [SV67: $i] :
( ( ( one_to_one @ SV67 )
= $false )
| ( ( function @ ( inverse @ SV67 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[637]) ).
thf(739,plain,
! [SV68: $i] :
( ( ( function @ ( inverse @ SV68 ) )
= $false )
| ( ( ~ ( function @ SV68 )
| ( one_to_one @ SV68 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[638]) ).
thf(740,plain,
! [SV71: $i] :
( ( ( operation @ SV71 )
= $false )
| ( ( function @ SV71 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[639]) ).
thf(741,plain,
! [SV72: $i] :
( ( ( operation @ SV72 )
= $false )
| ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV72 ) ) @ ( domain_of @ ( domain_of @ SV72 ) ) )
= ( domain_of @ SV72 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[640]) ).
thf(742,plain,
! [SV73: $i] :
( ( ( operation @ SV73 )
= $false )
| ( ( subclass @ ( range_of @ SV73 ) @ ( domain_of @ ( domain_of @ SV73 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[641]) ).
thf(743,plain,
! [SV74: $i] :
( ( ( function @ SV74 )
= $false )
| ( ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV74 ) ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
!= ( domain_of @ SV74 ) )
| ~ ( subclass @ ( range_of @ SV74 ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
| ( operation @ SV74 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[642]) ).
thf(744,plain,
! [SV205: $i,SV75: $i] :
( ( ( ! [SY451: $i] :
~ ( compatible @ SV75 @ SV205 @ SY451 ) )
= $true )
| ( ( function @ SV75 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[643]) ).
thf(745,plain,
! [SV161: $i,SV76: $i] :
( ( ( ! [SY419: $i] :
~ ( compatible @ SV76 @ SV161 @ SY419 ) )
= $true )
| ( ( ( domain_of @ ( domain_of @ SV161 ) )
= ( domain_of @ SV76 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[644]) ).
thf(746,plain,
! [SV206: $i,SV162: $i,SV77: $i] :
( ( ~ ( compatible @ SV77 @ SV162 @ SV206 )
| ( subclass @ ( range_of @ SV77 ) @ ( domain_of @ ( domain_of @ SV206 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[645]) ).
thf(747,plain,
! [SV78: $i] :
( ( ( function @ SV78 )
= $false )
| ( ( ! [SY334: $i] :
( ( ( domain_of @ ( domain_of @ SY334 ) )
!= ( domain_of @ SV78 ) )
| ! [SY335: $i] :
( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SY335 ) ) )
| ( compatible @ SV78 @ SY334 @ SY335 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[646]) ).
thf(748,plain,
! [SV163: $i,SV79: $i] :
( ( ( ! [SY421: $i] :
~ ( homomorphism @ SV79 @ SV163 @ SY421 ) )
= $true )
| ( ( operation @ SV163 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[647]) ).
thf(749,plain,
! [SV207: $i,SV164: $i,SV80: $i] :
( ( ~ ( homomorphism @ SV80 @ SV164 @ SV207 )
| ( operation @ SV207 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[648]) ).
thf(750,plain,
! [SV208: $i,SV165: $i,SV81: $i] :
( ( ~ ( homomorphism @ SV81 @ SV165 @ SV208 )
| ( compatible @ SV81 @ SV165 @ SV208 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[649]) ).
thf(751,plain,
! [SV209: $i,SV166: $i,SV82: $i] :
( ( ~ ( homomorphism @ SV82 @ SV166 @ SV209 )
| ! [SY452: $i,SY453: $i] :
( ~ ( member @ ( ordered_pair @ SY452 @ SY453 ) @ ( domain_of @ SV166 ) )
| ( ( apply @ SV209 @ ( ordered_pair @ ( apply @ SV82 @ SY452 ) @ ( apply @ SV82 @ SY453 ) ) )
= ( apply @ SV82 @ ( apply @ SV166 @ ( ordered_pair @ SY452 @ SY453 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[650]) ).
thf(752,plain,
! [SV83: $i] :
( ( ( operation @ SV83 )
= $false )
| ( ( ! [SY346: $i] :
( ~ ( operation @ SY346 )
| ! [SY347: $i] :
( ~ ( compatible @ SY347 @ SV83 @ SY346 )
| ( homomorphism @ SY347 @ SV83 @ SY346 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SY347 @ SV83 @ SY346 ) @ ( not_homomorphism2 @ SY347 @ SV83 @ SY346 ) ) @ ( domain_of @ SV83 ) ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[651]) ).
thf(753,plain,
! [SV84: $i] :
( ( ( operation @ SV84 )
= $false )
| ( ( ! [SY348: $i] :
( ~ ( operation @ SY348 )
| ! [SY349: $i] :
( ~ ( compatible @ SY349 @ SV84 @ SY348 )
| ( ( apply @ SY348 @ ( ordered_pair @ ( apply @ SY349 @ ( not_homomorphism1 @ SY349 @ SV84 @ SY348 ) ) @ ( apply @ SY349 @ ( not_homomorphism2 @ SY349 @ SV84 @ SY348 ) ) ) )
!= ( apply @ SY349 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SY349 @ SV84 @ SY348 ) @ ( not_homomorphism2 @ SY349 @ SV84 @ SY348 ) ) ) ) )
| ( homomorphism @ SY349 @ SV84 @ SY348 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[652]) ).
thf(754,plain,
! [SV167: $i,SV85: $i] :
( ( ( ! [SY427: $i,SY428: $i] :
~ ( member @ ( ordered_pair @ SV85 @ SV167 ) @ ( cross_product @ SY427 @ SY428 ) ) )
= $true )
| ( ( member @ SV85 @ ( unordered_pair @ SV85 @ SV167 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[653]) ).
thf(755,plain,
! [SV168: $i,SV86: $i] :
( ( ( ! [SY429: $i,SY430: $i] :
~ ( member @ ( ordered_pair @ SV86 @ SV168 ) @ ( cross_product @ SY429 @ SY430 ) ) )
= $true )
| ( ( member @ SV168 @ ( unordered_pair @ SV86 @ SV168 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[654]) ).
thf(756,plain,
! [SV169: $i,SV87: $i] :
( ( ( ! [SY431: $i,SY432: $i] :
~ ( member @ ( ordered_pair @ SV87 @ SV169 ) @ ( cross_product @ SY431 @ SY432 ) ) )
= $true )
| ( ( member @ SV87 @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[655]) ).
thf(757,plain,
! [SV170: $i,SV88: $i] :
( ( ( ! [SY433: $i,SY434: $i] :
~ ( member @ ( ordered_pair @ SV88 @ SV170 ) @ ( cross_product @ SY433 @ SY434 ) ) )
= $true )
| ( ( member @ SV170 @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[656]) ).
thf(758,plain,
! [SV171: $i,SV90: $i] :
( ( ( ~ ( subclass @ SV90 @ SV171 ) )
= $true )
| ( ( ! [SY435: $i] :
( ~ ( subclass @ SV171 @ SY435 )
| ( subclass @ SV90 @ SY435 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[657]) ).
thf(759,plain,
! [SV172: $i,SV91: $i] :
( ( ( SV91 = SV172 )
= $true )
| ( ( ( member @ ( not_subclass_element @ SV91 @ SV172 ) @ SV91 )
| ( member @ ( not_subclass_element @ SV172 @ SV91 ) @ SV172 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[658]) ).
thf(760,plain,
! [SV173: $i,SV92: $i] :
( ( ( ~ ( member @ ( not_subclass_element @ SV92 @ SV173 ) @ SV173 ) )
= $true )
| ( ( ( SV92 = SV173 )
| ( member @ ( not_subclass_element @ SV173 @ SV92 ) @ SV173 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[659]) ).
thf(761,plain,
! [SV174: $i,SV93: $i] :
( ( ( ~ ( member @ ( not_subclass_element @ SV93 @ SV174 ) @ SV174 ) )
= $true )
| ( ( ( SV174 = SV93 )
| ( member @ ( not_subclass_element @ SV174 @ SV93 ) @ SV174 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[660]) ).
thf(762,plain,
! [SV175: $i,SV94: $i] :
( ( ( ~ ( member @ ( not_subclass_element @ SV94 @ SV175 ) @ SV175 ) )
= $true )
| ( ( ~ ( member @ ( not_subclass_element @ SV175 @ SV94 ) @ SV94 )
| ( SV94 = SV175 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[661]) ).
thf(763,plain,
! [SV176: $i,SV95: $i] :
( ( member @ SV95 @ ( intersection @ ( complement @ SV176 ) @ SV176 ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[662]) ).
thf(764,plain,
! [SV98: $i] :
( ( ( subclass @ SV98 @ null_class )
= $false )
| ( ( SV98 = null_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[664]) ).
thf(765,plain,
! [SV103: $i,SV210: $i] :
( ( ( ( unordered_pair @ SV210 @ SV103 )
= ( singleton @ SV210 ) )
= $true )
| ( ( member @ SV103 @ universal_class )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[669]) ).
thf(766,plain,
! [SV211: $i,SV104: $i] :
( ( ( ( unordered_pair @ SV104 @ SV211 )
= ( singleton @ SV211 ) )
= $true )
| ( ( member @ SV104 @ universal_class )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[670]) ).
thf(767,plain,
! [SV180: $i,SV105: $i] :
( ( ( ( unordered_pair @ SV105 @ SV180 )
= null_class )
= $true )
| ( ( ( member @ SV105 @ universal_class )
| ( member @ SV180 @ universal_class ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[671]) ).
thf(768,plain,
! [SV212: $i,SV181: $i,SV106: $i] :
( ( ( ( unordered_pair @ SV106 @ SV181 )
!= ( unordered_pair @ SV106 @ SV212 ) )
| ~ ( member @ ( ordered_pair @ SV181 @ SV212 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV181 = SV212 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[672]) ).
thf(769,plain,
! [SV213: $i,SV182: $i,SV107: $i] :
( ( ( ( unordered_pair @ SV107 @ SV182 )
!= ( unordered_pair @ SV213 @ SV182 ) )
| ~ ( member @ ( ordered_pair @ SV107 @ SV213 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV107 = SV213 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[673]) ).
thf(770,plain,
! [SV108: $i] :
( ( ( member @ SV108 @ universal_class )
= $false )
| ( ( ! [SY379: $i] :
( ( unordered_pair @ SV108 @ SY379 )
!= null_class ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[674]) ).
thf(771,plain,
! [SV109: $i] :
( ( ( member @ SV109 @ universal_class )
= $false )
| ( ( ! [SY380: $i] :
( ( unordered_pair @ SY380 @ SV109 )
!= null_class ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[675]) ).
thf(772,plain,
! [SV183: $i,SV110: $i] :
( ( ( ! [SY438: $i,SY439: $i] :
~ ( member @ ( ordered_pair @ SV110 @ SV183 ) @ ( cross_product @ SY438 @ SY439 ) ) )
= $true )
| ( ( ( unordered_pair @ SV110 @ SV183 )
!= null_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[676]) ).
thf(773,plain,
! [SV184: $i,SV111: $i] :
( ( ( ~ ( member @ SV111 @ SV184 ) )
= $true )
| ( ( ! [SY440: $i] :
( ~ ( member @ SY440 @ SV184 )
| ( subclass @ ( unordered_pair @ SV111 @ SY440 ) @ SV184 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[677]) ).
thf(774,plain,
! [SV114: $i] :
( ( ( member @ SV114 @ universal_class )
= $false )
| ( ( member @ SV114 @ ( singleton @ SV114 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[679]) ).
thf(775,plain,
! [SV115: $i] :
( ( ( member @ SV115 @ universal_class )
= $false )
| ( ( ( singleton @ SV115 )
!= null_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[680]) ).
thf(776,plain,
! [SV186: $i,SV116: $i] :
( ( ( ~ ( member @ SV116 @ ( singleton @ SV186 ) ) )
= $true )
| ( ( SV116 = SV186 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[681]) ).
thf(777,plain,
! [SV187: $i,SV118: $i] :
( ( ( ( singleton @ SV118 )
!= ( singleton @ SV187 ) )
= $true )
| ( ( ~ ( member @ SV118 @ universal_class )
| ( SV118 = SV187 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[683]) ).
thf(778,plain,
! [SV188: $i,SV119: $i] :
( ( ( ( singleton @ SV119 )
!= ( singleton @ SV188 ) )
= $true )
| ( ( ~ ( member @ SV188 @ universal_class )
| ( SV119 = SV188 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[684]) ).
thf(779,plain,
! [SV214: $i,SV189: $i,SV120: $i] :
( ( ( ( unordered_pair @ SV120 @ SV189 )
!= ( singleton @ SV214 ) )
| ~ ( member @ SV214 @ universal_class )
| ( SV214 = SV120 )
| ( SV214 = SV189 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[685]) ).
thf(780,plain,
! [SV121: $i] :
( ( ( member @ SV121 @ universal_class )
= $false )
| ( ( member @ ( member_of @ ( singleton @ SV121 ) ) @ universal_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[686]) ).
thf(781,plain,
! [SV122: $i] :
( ( ( member @ SV122 @ universal_class )
= $false )
| ( ( ( singleton @ ( member_of @ ( singleton @ SV122 ) ) )
= ( singleton @ SV122 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[687]) ).
thf(782,plain,
! [SV125: $i,SV1: $i] :
( ( ( subclass @ SV1 @ SV125 )
= $false )
| ( ( ! [SY392: $i] :
( ~ ( member @ SY392 @ SV1 )
| ( member @ SY392 @ SV125 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[690]) ).
thf(783,plain,
! [SV127: $i,SV3: $i] :
( ( ( member @ ( not_subclass_element @ SV3 @ SV127 ) @ SV127 )
= $false )
| ( ( subclass @ SV3 @ SV127 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[692]) ).
thf(784,plain,
! [SV128: $i,SV5: $i] :
( ( ( SV5 = SV128 )
= $false )
| ( ( subclass @ SV5 @ SV128 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[693]) ).
thf(785,plain,
! [SV129: $i,SV6: $i] :
( ( ( SV6 = SV129 )
= $false )
| ( ( subclass @ SV129 @ SV6 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[694]) ).
thf(786,plain,
! [SV130: $i,SV7: $i] :
( ( ( subclass @ SV7 @ SV130 )
= $false )
| ( ( ~ ( subclass @ SV130 @ SV7 )
| ( SV7 = SV130 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[695]) ).
thf(787,plain,
! [SV190: $i,SV131: $i,SV8: $i] :
( ( ( ~ ( member @ SV8 @ ( unordered_pair @ SV131 @ SV190 ) ) )
= $true )
| ( ( ( SV8 = SV131 )
| ( SV8 = SV190 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[696]) ).
thf(788,plain,
! [SV215: $i,SV9: $i] :
( ( ( member @ SV9 @ ( unordered_pair @ SV9 @ SV215 ) )
= $true )
| ( ( member @ SV9 @ universal_class )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[697]) ).
thf(789,plain,
! [SV216: $i,SV10: $i] :
( ( ( member @ SV10 @ ( unordered_pair @ SV216 @ SV10 ) )
= $true )
| ( ( member @ SV10 @ universal_class )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[698]) ).
thf(790,plain,
! [SV191: $i,SV134: $i,SV14: $i] :
( ( ( ! [SY442: $i] :
~ ( member @ ( ordered_pair @ SV14 @ SV134 ) @ ( cross_product @ SV191 @ SY442 ) ) )
= $true )
| ( ( member @ SV14 @ SV191 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[699]) ).
thf(791,plain,
! [SV217: $i,SV192: $i,SV135: $i,SV15: $i] :
( ( ~ ( member @ ( ordered_pair @ SV15 @ SV135 ) @ ( cross_product @ SV192 @ SV217 ) )
| ( member @ SV135 @ SV217 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[700]) ).
thf(792,plain,
! [SV136: $i,SV16: $i] :
( ( ( member @ SV16 @ SV136 )
= $false )
| ( ( ! [SY398: $i,SY399: $i] :
( ~ ( member @ SY398 @ SY399 )
| ( member @ ( ordered_pair @ SV16 @ SY398 ) @ ( cross_product @ SV136 @ SY399 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[701]) ).
thf(793,plain,
! [SV218: $i,SV193: $i,SV17: $i] :
( ( ( ~ ( member @ SV17 @ ( cross_product @ SV193 @ SV218 ) ) )
= $true )
| ( ( ( ordered_pair @ ( first @ SV17 ) @ ( second @ SV17 ) )
= SV17 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[702]) ).
thf(794,plain,
! [SV137: $i,SV18: $i] :
( ( ( member @ ( ordered_pair @ SV18 @ SV137 ) @ element_relation )
= $false )
| ( ( member @ SV18 @ SV137 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[703]) ).
thf(795,plain,
! [SV138: $i,SV19: $i] :
( ( ( member @ ( ordered_pair @ SV19 @ SV138 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( ~ ( member @ SV19 @ SV138 )
| ( member @ ( ordered_pair @ SV19 @ SV138 ) @ element_relation ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[704]) ).
thf(796,plain,
! [SV219: $i,SV139: $i,SV20: $i] :
( ( ( ~ ( member @ SV20 @ ( intersection @ SV139 @ SV219 ) ) )
= $true )
| ( ( member @ SV20 @ SV139 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[705]) ).
thf(797,plain,
! [SV194: $i,SV140: $i,SV21: $i] :
( ( ( ~ ( member @ SV21 @ ( intersection @ SV140 @ SV194 ) ) )
= $true )
| ( ( member @ SV21 @ SV194 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[706]) ).
thf(798,plain,
! [SV141: $i,SV22: $i] :
( ( ( member @ SV22 @ SV141 )
= $false )
| ( ( ! [SY402: $i] :
( ~ ( member @ SV22 @ SY402 )
| ( member @ SV22 @ ( intersection @ SV141 @ SY402 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[707]) ).
thf(799,plain,
! [SV142: $i,SV23: $i] :
( ( ( member @ SV23 @ ( complement @ SV142 ) )
= $false )
| ( ( ~ ( member @ SV23 @ SV142 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[708]) ).
thf(800,plain,
! [SV220: $i,SV24: $i] :
( ( ( ( member @ SV24 @ ( complement @ SV220 ) )
| ( member @ SV24 @ SV220 ) )
= $true )
| ( ( member @ SV24 @ universal_class )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[709]) ).
thf(801,plain,
! [SV147: $i,SV29: $i] :
( ( ( ( restrict @ SV29 @ ( singleton @ SV147 ) @ universal_class )
= null_class )
= $false )
| ( ( ~ ( member @ SV147 @ ( domain_of @ SV29 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[712]) ).
thf(802,plain,
! [SV30: $i,SV221: $i] :
( ( ( ( ( restrict @ SV221 @ ( singleton @ SV30 ) @ universal_class )
= null_class )
| ( member @ SV30 @ ( domain_of @ SV221 ) ) )
= $true )
| ( ( member @ SV30 @ universal_class )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[713]) ).
thf(803,plain,
! [SV222: $i,SV197: $i,SV148: $i,SV32: $i] :
( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV148 ) @ SV197 ) @ ( rotate @ SV222 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV148 @ SV197 ) @ SV32 ) @ SV222 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[714]) ).
thf(804,plain,
! [SV223: $i,SV198: $i,SV149: $i,SV33: $i] :
( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV149 ) @ SV198 ) @ SV223 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( rotate @ SV223 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[715]) ).
thf(805,plain,
! [SV224: $i,SV199: $i,SV150: $i,SV35: $i] :
( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV35 @ SV150 ) @ SV199 ) @ ( flip @ SV224 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV150 @ SV35 ) @ SV199 ) @ SV224 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[716]) ).
thf(806,plain,
! [SV225: $i,SV200: $i,SV151: $i,SV36: $i] :
( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV36 @ SV151 ) @ SV200 ) @ SV225 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( flip @ SV225 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[717]) ).
thf(807,plain,
! [SV155: $i,SV43: $i] :
( ( ( member @ ( ordered_pair @ SV43 @ SV155 ) @ successor_relation )
= $false )
| ( ( ( successor @ SV43 )
= SV155 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[720]) ).
thf(808,plain,
! [SV156: $i,SV44: $i] :
( ( ( ( successor @ SV44 )
= SV156 )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ SV44 @ SV156 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV44 @ SV156 ) @ successor_relation ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[721]) ).
thf(809,plain,
! [SV47: $i] :
( ( ( ~ ( subclass @ ( image @ successor_relation @ SV47 ) @ SV47 ) )
= $true )
| ( ( inductive @ SV47 )
= $true )
| ( ( member @ null_class @ SV47 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[724]) ).
thf(810,plain,
! [SV226: $i,SV203: $i,SV158: $i,SV54: $i] :
( ( ~ ( member @ ( ordered_pair @ SV54 @ SV158 ) @ ( compose @ SV203 @ SV226 ) )
| ( member @ SV158 @ ( image @ SV203 @ ( image @ SV226 @ ( singleton @ SV54 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[728]) ).
thf(811,plain,
! [SV227: $i,SV204: $i,SV159: $i,SV55: $i] :
( ( ~ ( member @ SV55 @ ( image @ SV159 @ ( image @ SV204 @ ( singleton @ SV227 ) ) ) )
| ~ ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( compose @ SV159 @ SV204 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[729]) ).
thf(812,plain,
! [SV60: $i] :
( ( ( ~ ( subclass @ ( compose @ SV60 @ ( inverse @ SV60 ) ) @ identity_relation ) )
= $true )
| ( ( function @ SV60 )
= $true )
| ( ( subclass @ SV60 @ ( cross_product @ universal_class @ universal_class ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[734]) ).
thf(813,plain,
! [SV61: $i,SV228: $i] :
( ( ( ~ ( member @ SV228 @ universal_class )
| ( member @ ( image @ SV61 @ SV228 ) @ universal_class ) )
= $true )
| ( ( function @ SV61 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[735]) ).
thf(814,plain,
! [SV65: $i] :
( ( ( SV65 = null_class )
= $true )
| ( ( member @ ( apply @ choice @ SV65 ) @ SV65 )
= $true )
| ( ( member @ SV65 @ universal_class )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[736]) ).
thf(815,plain,
! [SV68: $i] :
( ( ( ~ ( function @ SV68 ) )
= $true )
| ( ( one_to_one @ SV68 )
= $true )
| ( ( function @ ( inverse @ SV68 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[739]) ).
thf(816,plain,
! [SV74: $i] :
( ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV74 ) ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
!= ( domain_of @ SV74 ) )
= $true )
| ( ( ~ ( subclass @ ( range_of @ SV74 ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
| ( operation @ SV74 ) )
= $true )
| ( ( function @ SV74 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[743]) ).
thf(817,plain,
! [SV229: $i,SV205: $i,SV75: $i] :
( ( ( ~ ( compatible @ SV75 @ SV205 @ SV229 ) )
= $true )
| ( ( function @ SV75 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[744]) ).
thf(818,plain,
! [SV230: $i,SV161: $i,SV76: $i] :
( ( ( ~ ( compatible @ SV76 @ SV161 @ SV230 ) )
= $true )
| ( ( ( domain_of @ ( domain_of @ SV161 ) )
= ( domain_of @ SV76 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[745]) ).
thf(819,plain,
! [SV206: $i,SV162: $i,SV77: $i] :
( ( ( ~ ( compatible @ SV77 @ SV162 @ SV206 ) )
= $true )
| ( ( subclass @ ( range_of @ SV77 ) @ ( domain_of @ ( domain_of @ SV206 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[746]) ).
thf(820,plain,
! [SV78: $i,SV231: $i] :
( ( ( ( ( domain_of @ ( domain_of @ SV231 ) )
!= ( domain_of @ SV78 ) )
| ! [SY454: $i] :
( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SY454 ) ) )
| ( compatible @ SV78 @ SV231 @ SY454 ) ) )
= $true )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[747]) ).
thf(821,plain,
! [SV232: $i,SV163: $i,SV79: $i] :
( ( ( ~ ( homomorphism @ SV79 @ SV163 @ SV232 ) )
= $true )
| ( ( operation @ SV163 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[748]) ).
thf(822,plain,
! [SV207: $i,SV164: $i,SV80: $i] :
( ( ( ~ ( homomorphism @ SV80 @ SV164 @ SV207 ) )
= $true )
| ( ( operation @ SV207 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[749]) ).
thf(823,plain,
! [SV208: $i,SV165: $i,SV81: $i] :
( ( ( ~ ( homomorphism @ SV81 @ SV165 @ SV208 ) )
= $true )
| ( ( compatible @ SV81 @ SV165 @ SV208 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[750]) ).
thf(824,plain,
! [SV209: $i,SV166: $i,SV82: $i] :
( ( ( ~ ( homomorphism @ SV82 @ SV166 @ SV209 ) )
= $true )
| ( ( ! [SY452: $i,SY453: $i] :
( ~ ( member @ ( ordered_pair @ SY452 @ SY453 ) @ ( domain_of @ SV166 ) )
| ( ( apply @ SV209 @ ( ordered_pair @ ( apply @ SV82 @ SY452 ) @ ( apply @ SV82 @ SY453 ) ) )
= ( apply @ SV82 @ ( apply @ SV166 @ ( ordered_pair @ SY452 @ SY453 ) ) ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[751]) ).
thf(825,plain,
! [SV83: $i,SV233: $i] :
( ( ( ~ ( operation @ SV233 )
| ! [SY455: $i] :
( ~ ( compatible @ SY455 @ SV83 @ SV233 )
| ( homomorphism @ SY455 @ SV83 @ SV233 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SY455 @ SV83 @ SV233 ) @ ( not_homomorphism2 @ SY455 @ SV83 @ SV233 ) ) @ ( domain_of @ SV83 ) ) ) )
= $true )
| ( ( operation @ SV83 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[752]) ).
thf(826,plain,
! [SV84: $i,SV234: $i] :
( ( ( ~ ( operation @ SV234 )
| ! [SY456: $i] :
( ~ ( compatible @ SY456 @ SV84 @ SV234 )
| ( ( apply @ SV234 @ ( ordered_pair @ ( apply @ SY456 @ ( not_homomorphism1 @ SY456 @ SV84 @ SV234 ) ) @ ( apply @ SY456 @ ( not_homomorphism2 @ SY456 @ SV84 @ SV234 ) ) ) )
!= ( apply @ SY456 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SY456 @ SV84 @ SV234 ) @ ( not_homomorphism2 @ SY456 @ SV84 @ SV234 ) ) ) ) )
| ( homomorphism @ SY456 @ SV84 @ SV234 ) ) )
= $true )
| ( ( operation @ SV84 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[753]) ).
thf(827,plain,
! [SV235: $i,SV167: $i,SV85: $i] :
( ( ( ! [SY457: $i] :
~ ( member @ ( ordered_pair @ SV85 @ SV167 ) @ ( cross_product @ SV235 @ SY457 ) ) )
= $true )
| ( ( member @ SV85 @ ( unordered_pair @ SV85 @ SV167 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[754]) ).
thf(828,plain,
! [SV236: $i,SV168: $i,SV86: $i] :
( ( ( ! [SY458: $i] :
~ ( member @ ( ordered_pair @ SV86 @ SV168 ) @ ( cross_product @ SV236 @ SY458 ) ) )
= $true )
| ( ( member @ SV168 @ ( unordered_pair @ SV86 @ SV168 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[755]) ).
thf(829,plain,
! [SV237: $i,SV169: $i,SV87: $i] :
( ( ( ! [SY459: $i] :
~ ( member @ ( ordered_pair @ SV87 @ SV169 ) @ ( cross_product @ SV237 @ SY459 ) ) )
= $true )
| ( ( member @ SV87 @ universal_class )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[756]) ).
thf(830,plain,
! [SV238: $i,SV170: $i,SV88: $i] :
( ( ( ! [SY460: $i] :
~ ( member @ ( ordered_pair @ SV88 @ SV170 ) @ ( cross_product @ SV238 @ SY460 ) ) )
= $true )
| ( ( member @ SV170 @ universal_class )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[757]) ).
thf(831,plain,
! [SV171: $i,SV90: $i] :
( ( ( subclass @ SV90 @ SV171 )
= $false )
| ( ( ! [SY435: $i] :
( ~ ( subclass @ SV171 @ SY435 )
| ( subclass @ SV90 @ SY435 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[758]) ).
thf(832,plain,
! [SV172: $i,SV91: $i] :
( ( ( member @ ( not_subclass_element @ SV91 @ SV172 ) @ SV91 )
= $true )
| ( ( member @ ( not_subclass_element @ SV172 @ SV91 ) @ SV172 )
= $true )
| ( ( SV91 = SV172 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[759]) ).
thf(833,plain,
! [SV173: $i,SV92: $i] :
( ( ( member @ ( not_subclass_element @ SV92 @ SV173 ) @ SV173 )
= $false )
| ( ( ( SV92 = SV173 )
| ( member @ ( not_subclass_element @ SV173 @ SV92 ) @ SV173 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[760]) ).
thf(834,plain,
! [SV174: $i,SV93: $i] :
( ( ( member @ ( not_subclass_element @ SV93 @ SV174 ) @ SV174 )
= $false )
| ( ( ( SV174 = SV93 )
| ( member @ ( not_subclass_element @ SV174 @ SV93 ) @ SV174 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[761]) ).
thf(835,plain,
! [SV175: $i,SV94: $i] :
( ( ( member @ ( not_subclass_element @ SV94 @ SV175 ) @ SV175 )
= $false )
| ( ( ~ ( member @ ( not_subclass_element @ SV175 @ SV94 ) @ SV94 )
| ( SV94 = SV175 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[762]) ).
thf(836,plain,
! [SV180: $i,SV105: $i] :
( ( ( member @ SV105 @ universal_class )
= $true )
| ( ( member @ SV180 @ universal_class )
= $true )
| ( ( ( unordered_pair @ SV105 @ SV180 )
= null_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[767]) ).
thf(837,plain,
! [SV212: $i,SV181: $i,SV106: $i] :
( ( ( ( unordered_pair @ SV106 @ SV181 )
!= ( unordered_pair @ SV106 @ SV212 ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ SV181 @ SV212 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV181 = SV212 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[768]) ).
thf(838,plain,
! [SV213: $i,SV182: $i,SV107: $i] :
( ( ( ( unordered_pair @ SV107 @ SV182 )
!= ( unordered_pair @ SV213 @ SV182 ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ SV107 @ SV213 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV107 = SV213 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[769]) ).
thf(839,plain,
! [SV239: $i,SV108: $i] :
( ( ( ( unordered_pair @ SV108 @ SV239 )
!= null_class )
= $true )
| ( ( member @ SV108 @ universal_class )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[770]) ).
thf(840,plain,
! [SV109: $i,SV240: $i] :
( ( ( ( unordered_pair @ SV240 @ SV109 )
!= null_class )
= $true )
| ( ( member @ SV109 @ universal_class )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[771]) ).
thf(841,plain,
! [SV241: $i,SV183: $i,SV110: $i] :
( ( ( ! [SY461: $i] :
~ ( member @ ( ordered_pair @ SV110 @ SV183 ) @ ( cross_product @ SV241 @ SY461 ) ) )
= $true )
| ( ( ( unordered_pair @ SV110 @ SV183 )
!= null_class )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[772]) ).
thf(842,plain,
! [SV184: $i,SV111: $i] :
( ( ( member @ SV111 @ SV184 )
= $false )
| ( ( ! [SY440: $i] :
( ~ ( member @ SY440 @ SV184 )
| ( subclass @ ( unordered_pair @ SV111 @ SY440 ) @ SV184 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[773]) ).
thf(843,plain,
! [SV115: $i] :
( ( ( ( singleton @ SV115 )
= null_class )
= $false )
| ( ( member @ SV115 @ universal_class )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[775]) ).
thf(844,plain,
! [SV186: $i,SV116: $i] :
( ( ( member @ SV116 @ ( singleton @ SV186 ) )
= $false )
| ( ( SV116 = SV186 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[776]) ).
thf(845,plain,
! [SV187: $i,SV118: $i] :
( ( ( ( singleton @ SV118 )
= ( singleton @ SV187 ) )
= $false )
| ( ( ~ ( member @ SV118 @ universal_class )
| ( SV118 = SV187 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[777]) ).
thf(846,plain,
! [SV188: $i,SV119: $i] :
( ( ( ( singleton @ SV119 )
= ( singleton @ SV188 ) )
= $false )
| ( ( ~ ( member @ SV188 @ universal_class )
| ( SV119 = SV188 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[778]) ).
thf(847,plain,
! [SV214: $i,SV189: $i,SV120: $i] :
( ( ( ( unordered_pair @ SV120 @ SV189 )
!= ( singleton @ SV214 ) )
= $true )
| ( ( ~ ( member @ SV214 @ universal_class )
| ( SV214 = SV120 )
| ( SV214 = SV189 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[779]) ).
thf(848,plain,
! [SV125: $i,SV1: $i,SV242: $i] :
( ( ( ~ ( member @ SV242 @ SV1 )
| ( member @ SV242 @ SV125 ) )
= $true )
| ( ( subclass @ SV1 @ SV125 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[782]) ).
thf(849,plain,
! [SV7: $i,SV130: $i] :
( ( ( ~ ( subclass @ SV130 @ SV7 ) )
= $true )
| ( ( SV7 = SV130 )
= $true )
| ( ( subclass @ SV7 @ SV130 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[786]) ).
thf(850,plain,
! [SV190: $i,SV131: $i,SV8: $i] :
( ( ( member @ SV8 @ ( unordered_pair @ SV131 @ SV190 ) )
= $false )
| ( ( ( SV8 = SV131 )
| ( SV8 = SV190 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[787]) ).
thf(851,plain,
! [SV243: $i,SV191: $i,SV134: $i,SV14: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV14 @ SV134 ) @ ( cross_product @ SV191 @ SV243 ) ) )
= $true )
| ( ( member @ SV14 @ SV191 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[790]) ).
thf(852,plain,
! [SV217: $i,SV192: $i,SV135: $i,SV15: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV15 @ SV135 ) @ ( cross_product @ SV192 @ SV217 ) ) )
= $true )
| ( ( member @ SV135 @ SV217 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[791]) ).
thf(853,plain,
! [SV136: $i,SV16: $i,SV244: $i] :
( ( ( ! [SY462: $i] :
( ~ ( member @ SV244 @ SY462 )
| ( member @ ( ordered_pair @ SV16 @ SV244 ) @ ( cross_product @ SV136 @ SY462 ) ) ) )
= $true )
| ( ( member @ SV16 @ SV136 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[792]) ).
thf(854,plain,
! [SV218: $i,SV193: $i,SV17: $i] :
( ( ( member @ SV17 @ ( cross_product @ SV193 @ SV218 ) )
= $false )
| ( ( ( ordered_pair @ ( first @ SV17 ) @ ( second @ SV17 ) )
= SV17 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[793]) ).
thf(855,plain,
! [SV138: $i,SV19: $i] :
( ( ( ~ ( member @ SV19 @ SV138 ) )
= $true )
| ( ( member @ ( ordered_pair @ SV19 @ SV138 ) @ element_relation )
= $true )
| ( ( member @ ( ordered_pair @ SV19 @ SV138 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[795]) ).
thf(856,plain,
! [SV219: $i,SV139: $i,SV20: $i] :
( ( ( member @ SV20 @ ( intersection @ SV139 @ SV219 ) )
= $false )
| ( ( member @ SV20 @ SV139 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[796]) ).
thf(857,plain,
! [SV194: $i,SV140: $i,SV21: $i] :
( ( ( member @ SV21 @ ( intersection @ SV140 @ SV194 ) )
= $false )
| ( ( member @ SV21 @ SV194 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[797]) ).
thf(858,plain,
! [SV141: $i,SV245: $i,SV22: $i] :
( ( ( ~ ( member @ SV22 @ SV245 )
| ( member @ SV22 @ ( intersection @ SV141 @ SV245 ) ) )
= $true )
| ( ( member @ SV22 @ SV141 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[798]) ).
thf(859,plain,
! [SV142: $i,SV23: $i] :
( ( ( member @ SV23 @ SV142 )
= $false )
| ( ( member @ SV23 @ ( complement @ SV142 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[799]) ).
thf(860,plain,
! [SV220: $i,SV24: $i] :
( ( ( member @ SV24 @ ( complement @ SV220 ) )
= $true )
| ( ( member @ SV24 @ SV220 )
= $true )
| ( ( member @ SV24 @ universal_class )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[800]) ).
thf(861,plain,
! [SV29: $i,SV147: $i] :
( ( ( member @ SV147 @ ( domain_of @ SV29 ) )
= $false )
| ( ( ( restrict @ SV29 @ ( singleton @ SV147 ) @ universal_class )
= null_class )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[801]) ).
thf(862,plain,
! [SV30: $i,SV221: $i] :
( ( ( ( restrict @ SV221 @ ( singleton @ SV30 ) @ universal_class )
= null_class )
= $true )
| ( ( member @ SV30 @ ( domain_of @ SV221 ) )
= $true )
| ( ( member @ SV30 @ universal_class )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[802]) ).
thf(863,plain,
! [SV222: $i,SV197: $i,SV148: $i,SV32: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV148 ) @ SV197 ) @ ( rotate @ SV222 ) ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV148 @ SV197 ) @ SV32 ) @ SV222 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[803]) ).
thf(864,plain,
! [SV223: $i,SV198: $i,SV149: $i,SV33: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV149 ) @ SV198 ) @ SV223 ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( rotate @ SV223 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[804]) ).
thf(865,plain,
! [SV224: $i,SV199: $i,SV150: $i,SV35: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV35 @ SV150 ) @ SV199 ) @ ( flip @ SV224 ) ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV150 @ SV35 ) @ SV199 ) @ SV224 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[805]) ).
thf(866,plain,
! [SV225: $i,SV200: $i,SV151: $i,SV36: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV36 @ SV151 ) @ SV200 ) @ SV225 ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( flip @ SV225 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[806]) ).
thf(867,plain,
! [SV156: $i,SV44: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV44 @ SV156 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( member @ ( ordered_pair @ SV44 @ SV156 ) @ successor_relation )
= $true )
| ( ( ( successor @ SV44 )
= SV156 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[808]) ).
thf(868,plain,
! [SV47: $i] :
( ( ( subclass @ ( image @ successor_relation @ SV47 ) @ SV47 )
= $false )
| ( ( inductive @ SV47 )
= $true )
| ( ( member @ null_class @ SV47 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[809]) ).
thf(869,plain,
! [SV226: $i,SV203: $i,SV158: $i,SV54: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV54 @ SV158 ) @ ( compose @ SV203 @ SV226 ) ) )
= $true )
| ( ( member @ SV158 @ ( image @ SV203 @ ( image @ SV226 @ ( singleton @ SV54 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[810]) ).
thf(870,plain,
! [SV227: $i,SV204: $i,SV159: $i,SV55: $i] :
( ( ( ~ ( member @ SV55 @ ( image @ SV159 @ ( image @ SV204 @ ( singleton @ SV227 ) ) ) ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( compose @ SV159 @ SV204 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[811]) ).
thf(871,plain,
! [SV60: $i] :
( ( ( subclass @ ( compose @ SV60 @ ( inverse @ SV60 ) ) @ identity_relation )
= $false )
| ( ( function @ SV60 )
= $true )
| ( ( subclass @ SV60 @ ( cross_product @ universal_class @ universal_class ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[812]) ).
thf(872,plain,
! [SV61: $i,SV228: $i] :
( ( ( ~ ( member @ SV228 @ universal_class ) )
= $true )
| ( ( member @ ( image @ SV61 @ SV228 ) @ universal_class )
= $true )
| ( ( function @ SV61 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[813]) ).
thf(873,plain,
! [SV68: $i] :
( ( ( function @ SV68 )
= $false )
| ( ( one_to_one @ SV68 )
= $true )
| ( ( function @ ( inverse @ SV68 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[815]) ).
thf(874,plain,
! [SV74: $i] :
( ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV74 ) ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
= ( domain_of @ SV74 ) )
= $false )
| ( ( ~ ( subclass @ ( range_of @ SV74 ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
| ( operation @ SV74 ) )
= $true )
| ( ( function @ SV74 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[816]) ).
thf(875,plain,
! [SV229: $i,SV205: $i,SV75: $i] :
( ( ( compatible @ SV75 @ SV205 @ SV229 )
= $false )
| ( ( function @ SV75 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[817]) ).
thf(876,plain,
! [SV230: $i,SV161: $i,SV76: $i] :
( ( ( compatible @ SV76 @ SV161 @ SV230 )
= $false )
| ( ( ( domain_of @ ( domain_of @ SV161 ) )
= ( domain_of @ SV76 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[818]) ).
thf(877,plain,
! [SV206: $i,SV162: $i,SV77: $i] :
( ( ( compatible @ SV77 @ SV162 @ SV206 )
= $false )
| ( ( subclass @ ( range_of @ SV77 ) @ ( domain_of @ ( domain_of @ SV206 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[819]) ).
thf(878,plain,
! [SV78: $i,SV231: $i] :
( ( ( ( domain_of @ ( domain_of @ SV231 ) )
!= ( domain_of @ SV78 ) )
= $true )
| ( ( ! [SY454: $i] :
( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SY454 ) ) )
| ( compatible @ SV78 @ SV231 @ SY454 ) ) )
= $true )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[820]) ).
thf(879,plain,
! [SV232: $i,SV163: $i,SV79: $i] :
( ( ( homomorphism @ SV79 @ SV163 @ SV232 )
= $false )
| ( ( operation @ SV163 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[821]) ).
thf(880,plain,
! [SV207: $i,SV164: $i,SV80: $i] :
( ( ( homomorphism @ SV80 @ SV164 @ SV207 )
= $false )
| ( ( operation @ SV207 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[822]) ).
thf(881,plain,
! [SV208: $i,SV165: $i,SV81: $i] :
( ( ( homomorphism @ SV81 @ SV165 @ SV208 )
= $false )
| ( ( compatible @ SV81 @ SV165 @ SV208 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[823]) ).
thf(882,plain,
! [SV209: $i,SV166: $i,SV82: $i] :
( ( ( homomorphism @ SV82 @ SV166 @ SV209 )
= $false )
| ( ( ! [SY452: $i,SY453: $i] :
( ~ ( member @ ( ordered_pair @ SY452 @ SY453 ) @ ( domain_of @ SV166 ) )
| ( ( apply @ SV209 @ ( ordered_pair @ ( apply @ SV82 @ SY452 ) @ ( apply @ SV82 @ SY453 ) ) )
= ( apply @ SV82 @ ( apply @ SV166 @ ( ordered_pair @ SY452 @ SY453 ) ) ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[824]) ).
thf(883,plain,
! [SV83: $i,SV233: $i] :
( ( ( ~ ( operation @ SV233 ) )
= $true )
| ( ( ! [SY455: $i] :
( ~ ( compatible @ SY455 @ SV83 @ SV233 )
| ( homomorphism @ SY455 @ SV83 @ SV233 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SY455 @ SV83 @ SV233 ) @ ( not_homomorphism2 @ SY455 @ SV83 @ SV233 ) ) @ ( domain_of @ SV83 ) ) ) )
= $true )
| ( ( operation @ SV83 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[825]) ).
thf(884,plain,
! [SV84: $i,SV234: $i] :
( ( ( ~ ( operation @ SV234 ) )
= $true )
| ( ( ! [SY456: $i] :
( ~ ( compatible @ SY456 @ SV84 @ SV234 )
| ( ( apply @ SV234 @ ( ordered_pair @ ( apply @ SY456 @ ( not_homomorphism1 @ SY456 @ SV84 @ SV234 ) ) @ ( apply @ SY456 @ ( not_homomorphism2 @ SY456 @ SV84 @ SV234 ) ) ) )
!= ( apply @ SY456 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SY456 @ SV84 @ SV234 ) @ ( not_homomorphism2 @ SY456 @ SV84 @ SV234 ) ) ) ) )
| ( homomorphism @ SY456 @ SV84 @ SV234 ) ) )
= $true )
| ( ( operation @ SV84 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[826]) ).
thf(885,plain,
! [SV246: $i,SV235: $i,SV167: $i,SV85: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV85 @ SV167 ) @ ( cross_product @ SV235 @ SV246 ) ) )
= $true )
| ( ( member @ SV85 @ ( unordered_pair @ SV85 @ SV167 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[827]) ).
thf(886,plain,
! [SV247: $i,SV236: $i,SV168: $i,SV86: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV86 @ SV168 ) @ ( cross_product @ SV236 @ SV247 ) ) )
= $true )
| ( ( member @ SV168 @ ( unordered_pair @ SV86 @ SV168 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[828]) ).
thf(887,plain,
! [SV248: $i,SV237: $i,SV169: $i,SV87: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV87 @ SV169 ) @ ( cross_product @ SV237 @ SV248 ) ) )
= $true )
| ( ( member @ SV87 @ universal_class )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[829]) ).
thf(888,plain,
! [SV249: $i,SV238: $i,SV170: $i,SV88: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV88 @ SV170 ) @ ( cross_product @ SV238 @ SV249 ) ) )
= $true )
| ( ( member @ SV170 @ universal_class )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[830]) ).
thf(889,plain,
! [SV90: $i,SV250: $i,SV171: $i] :
( ( ( ~ ( subclass @ SV171 @ SV250 )
| ( subclass @ SV90 @ SV250 ) )
= $true )
| ( ( subclass @ SV90 @ SV171 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[831]) ).
thf(890,plain,
! [SV173: $i,SV92: $i] :
( ( ( SV92 = SV173 )
= $true )
| ( ( member @ ( not_subclass_element @ SV173 @ SV92 ) @ SV173 )
= $true )
| ( ( member @ ( not_subclass_element @ SV92 @ SV173 ) @ SV173 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[833]) ).
thf(891,plain,
! [SV93: $i,SV174: $i] :
( ( ( SV174 = SV93 )
= $true )
| ( ( member @ ( not_subclass_element @ SV174 @ SV93 ) @ SV174 )
= $true )
| ( ( member @ ( not_subclass_element @ SV93 @ SV174 ) @ SV174 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[834]) ).
thf(892,plain,
! [SV94: $i,SV175: $i] :
( ( ( ~ ( member @ ( not_subclass_element @ SV175 @ SV94 ) @ SV94 ) )
= $true )
| ( ( SV94 = SV175 )
= $true )
| ( ( member @ ( not_subclass_element @ SV94 @ SV175 ) @ SV175 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[835]) ).
thf(893,plain,
! [SV212: $i,SV181: $i,SV106: $i] :
( ( ( ( unordered_pair @ SV106 @ SV181 )
= ( unordered_pair @ SV106 @ SV212 ) )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ SV181 @ SV212 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV181 = SV212 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[837]) ).
thf(894,plain,
! [SV213: $i,SV182: $i,SV107: $i] :
( ( ( ( unordered_pair @ SV107 @ SV182 )
= ( unordered_pair @ SV213 @ SV182 ) )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ SV107 @ SV213 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( SV107 = SV213 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[838]) ).
thf(895,plain,
! [SV239: $i,SV108: $i] :
( ( ( ( unordered_pair @ SV108 @ SV239 )
= null_class )
= $false )
| ( ( member @ SV108 @ universal_class )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[839]) ).
thf(896,plain,
! [SV109: $i,SV240: $i] :
( ( ( ( unordered_pair @ SV240 @ SV109 )
= null_class )
= $false )
| ( ( member @ SV109 @ universal_class )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[840]) ).
thf(897,plain,
! [SV251: $i,SV241: $i,SV183: $i,SV110: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV110 @ SV183 ) @ ( cross_product @ SV241 @ SV251 ) ) )
= $true )
| ( ( ( unordered_pair @ SV110 @ SV183 )
!= null_class )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[841]) ).
thf(898,plain,
! [SV111: $i,SV184: $i,SV252: $i] :
( ( ( ~ ( member @ SV252 @ SV184 )
| ( subclass @ ( unordered_pair @ SV111 @ SV252 ) @ SV184 ) )
= $true )
| ( ( member @ SV111 @ SV184 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[842]) ).
thf(899,plain,
! [SV187: $i,SV118: $i] :
( ( ( ~ ( member @ SV118 @ universal_class ) )
= $true )
| ( ( SV118 = SV187 )
= $true )
| ( ( ( singleton @ SV118 )
= ( singleton @ SV187 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[845]) ).
thf(900,plain,
! [SV119: $i,SV188: $i] :
( ( ( ~ ( member @ SV188 @ universal_class ) )
= $true )
| ( ( SV119 = SV188 )
= $true )
| ( ( ( singleton @ SV119 )
= ( singleton @ SV188 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[846]) ).
thf(901,plain,
! [SV214: $i,SV189: $i,SV120: $i] :
( ( ( ( unordered_pair @ SV120 @ SV189 )
= ( singleton @ SV214 ) )
= $false )
| ( ( ~ ( member @ SV214 @ universal_class )
| ( SV214 = SV120 )
| ( SV214 = SV189 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[847]) ).
thf(902,plain,
! [SV125: $i,SV1: $i,SV242: $i] :
( ( ( ~ ( member @ SV242 @ SV1 ) )
= $true )
| ( ( member @ SV242 @ SV125 )
= $true )
| ( ( subclass @ SV1 @ SV125 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[848]) ).
thf(903,plain,
! [SV7: $i,SV130: $i] :
( ( ( subclass @ SV130 @ SV7 )
= $false )
| ( ( SV7 = SV130 )
= $true )
| ( ( subclass @ SV7 @ SV130 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[849]) ).
thf(904,plain,
! [SV190: $i,SV131: $i,SV8: $i] :
( ( ( SV8 = SV131 )
= $true )
| ( ( SV8 = SV190 )
= $true )
| ( ( member @ SV8 @ ( unordered_pair @ SV131 @ SV190 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[850]) ).
thf(905,plain,
! [SV243: $i,SV191: $i,SV134: $i,SV14: $i] :
( ( ( member @ ( ordered_pair @ SV14 @ SV134 ) @ ( cross_product @ SV191 @ SV243 ) )
= $false )
| ( ( member @ SV14 @ SV191 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[851]) ).
thf(906,plain,
! [SV217: $i,SV192: $i,SV135: $i,SV15: $i] :
( ( ( member @ ( ordered_pair @ SV15 @ SV135 ) @ ( cross_product @ SV192 @ SV217 ) )
= $false )
| ( ( member @ SV135 @ SV217 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[852]) ).
thf(907,plain,
! [SV136: $i,SV16: $i,SV253: $i,SV244: $i] :
( ( ( ~ ( member @ SV244 @ SV253 )
| ( member @ ( ordered_pair @ SV16 @ SV244 ) @ ( cross_product @ SV136 @ SV253 ) ) )
= $true )
| ( ( member @ SV16 @ SV136 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[853]) ).
thf(908,plain,
! [SV138: $i,SV19: $i] :
( ( ( member @ SV19 @ SV138 )
= $false )
| ( ( member @ ( ordered_pair @ SV19 @ SV138 ) @ element_relation )
= $true )
| ( ( member @ ( ordered_pair @ SV19 @ SV138 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[855]) ).
thf(909,plain,
! [SV141: $i,SV245: $i,SV22: $i] :
( ( ( ~ ( member @ SV22 @ SV245 ) )
= $true )
| ( ( member @ SV22 @ ( intersection @ SV141 @ SV245 ) )
= $true )
| ( ( member @ SV22 @ SV141 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[858]) ).
thf(910,plain,
! [SV222: $i,SV197: $i,SV148: $i,SV32: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV148 ) @ SV197 ) @ ( rotate @ SV222 ) )
= $false )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV148 @ SV197 ) @ SV32 ) @ SV222 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[863]) ).
thf(911,plain,
! [SV223: $i,SV198: $i,SV149: $i,SV33: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV149 ) @ SV198 ) @ SV223 )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( rotate @ SV223 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[864]) ).
thf(912,plain,
! [SV224: $i,SV199: $i,SV150: $i,SV35: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV35 @ SV150 ) @ SV199 ) @ ( flip @ SV224 ) )
= $false )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV150 @ SV35 ) @ SV199 ) @ SV224 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[865]) ).
thf(913,plain,
! [SV225: $i,SV200: $i,SV151: $i,SV36: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV36 @ SV151 ) @ SV200 ) @ SV225 )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( flip @ SV225 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[866]) ).
thf(914,plain,
! [SV156: $i,SV44: $i] :
( ( ( member @ ( ordered_pair @ SV44 @ SV156 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( member @ ( ordered_pair @ SV44 @ SV156 ) @ successor_relation )
= $true )
| ( ( ( successor @ SV44 )
= SV156 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[867]) ).
thf(915,plain,
! [SV226: $i,SV203: $i,SV158: $i,SV54: $i] :
( ( ( member @ ( ordered_pair @ SV54 @ SV158 ) @ ( compose @ SV203 @ SV226 ) )
= $false )
| ( ( member @ SV158 @ ( image @ SV203 @ ( image @ SV226 @ ( singleton @ SV54 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[869]) ).
thf(916,plain,
! [SV227: $i,SV204: $i,SV159: $i,SV55: $i] :
( ( ( member @ SV55 @ ( image @ SV159 @ ( image @ SV204 @ ( singleton @ SV227 ) ) ) )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( compose @ SV159 @ SV204 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[870]) ).
thf(917,plain,
! [SV61: $i,SV228: $i] :
( ( ( member @ SV228 @ universal_class )
= $false )
| ( ( member @ ( image @ SV61 @ SV228 ) @ universal_class )
= $true )
| ( ( function @ SV61 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[872]) ).
thf(918,plain,
! [SV74: $i] :
( ( ( ~ ( subclass @ ( range_of @ SV74 ) @ ( domain_of @ ( domain_of @ SV74 ) ) ) )
= $true )
| ( ( operation @ SV74 )
= $true )
| ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV74 ) ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
= ( domain_of @ SV74 ) )
= $false )
| ( ( function @ SV74 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[874]) ).
thf(919,plain,
! [SV78: $i,SV231: $i] :
( ( ( ( domain_of @ ( domain_of @ SV231 ) )
= ( domain_of @ SV78 ) )
= $false )
| ( ( ! [SY454: $i] :
( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SY454 ) ) )
| ( compatible @ SV78 @ SV231 @ SY454 ) ) )
= $true )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[878]) ).
thf(920,plain,
! [SV82: $i,SV209: $i,SV166: $i,SV254: $i] :
( ( ( ! [SY463: $i] :
( ~ ( member @ ( ordered_pair @ SV254 @ SY463 ) @ ( domain_of @ SV166 ) )
| ( ( apply @ SV209 @ ( ordered_pair @ ( apply @ SV82 @ SV254 ) @ ( apply @ SV82 @ SY463 ) ) )
= ( apply @ SV82 @ ( apply @ SV166 @ ( ordered_pair @ SV254 @ SY463 ) ) ) ) ) )
= $true )
| ( ( homomorphism @ SV82 @ SV166 @ SV209 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[882]) ).
thf(921,plain,
! [SV83: $i,SV233: $i] :
( ( ( operation @ SV233 )
= $false )
| ( ( ! [SY455: $i] :
( ~ ( compatible @ SY455 @ SV83 @ SV233 )
| ( homomorphism @ SY455 @ SV83 @ SV233 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SY455 @ SV83 @ SV233 ) @ ( not_homomorphism2 @ SY455 @ SV83 @ SV233 ) ) @ ( domain_of @ SV83 ) ) ) )
= $true )
| ( ( operation @ SV83 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[883]) ).
thf(922,plain,
! [SV84: $i,SV234: $i] :
( ( ( operation @ SV234 )
= $false )
| ( ( ! [SY456: $i] :
( ~ ( compatible @ SY456 @ SV84 @ SV234 )
| ( ( apply @ SV234 @ ( ordered_pair @ ( apply @ SY456 @ ( not_homomorphism1 @ SY456 @ SV84 @ SV234 ) ) @ ( apply @ SY456 @ ( not_homomorphism2 @ SY456 @ SV84 @ SV234 ) ) ) )
!= ( apply @ SY456 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SY456 @ SV84 @ SV234 ) @ ( not_homomorphism2 @ SY456 @ SV84 @ SV234 ) ) ) ) )
| ( homomorphism @ SY456 @ SV84 @ SV234 ) ) )
= $true )
| ( ( operation @ SV84 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[884]) ).
thf(923,plain,
! [SV246: $i,SV235: $i,SV167: $i,SV85: $i] :
( ( ( member @ ( ordered_pair @ SV85 @ SV167 ) @ ( cross_product @ SV235 @ SV246 ) )
= $false )
| ( ( member @ SV85 @ ( unordered_pair @ SV85 @ SV167 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[885]) ).
thf(924,plain,
! [SV247: $i,SV236: $i,SV168: $i,SV86: $i] :
( ( ( member @ ( ordered_pair @ SV86 @ SV168 ) @ ( cross_product @ SV236 @ SV247 ) )
= $false )
| ( ( member @ SV168 @ ( unordered_pair @ SV86 @ SV168 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[886]) ).
thf(925,plain,
! [SV248: $i,SV237: $i,SV169: $i,SV87: $i] :
( ( ( member @ ( ordered_pair @ SV87 @ SV169 ) @ ( cross_product @ SV237 @ SV248 ) )
= $false )
| ( ( member @ SV87 @ universal_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[887]) ).
thf(926,plain,
! [SV249: $i,SV238: $i,SV170: $i,SV88: $i] :
( ( ( member @ ( ordered_pair @ SV88 @ SV170 ) @ ( cross_product @ SV238 @ SV249 ) )
= $false )
| ( ( member @ SV170 @ universal_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[888]) ).
thf(927,plain,
! [SV90: $i,SV250: $i,SV171: $i] :
( ( ( ~ ( subclass @ SV171 @ SV250 ) )
= $true )
| ( ( subclass @ SV90 @ SV250 )
= $true )
| ( ( subclass @ SV90 @ SV171 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[889]) ).
thf(928,plain,
! [SV94: $i,SV175: $i] :
( ( ( member @ ( not_subclass_element @ SV175 @ SV94 ) @ SV94 )
= $false )
| ( ( SV94 = SV175 )
= $true )
| ( ( member @ ( not_subclass_element @ SV94 @ SV175 ) @ SV175 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[892]) ).
thf(929,plain,
! [SV106: $i,SV212: $i,SV181: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV181 @ SV212 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( SV181 = SV212 )
= $true )
| ( ( ( unordered_pair @ SV106 @ SV181 )
= ( unordered_pair @ SV106 @ SV212 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[893]) ).
thf(930,plain,
! [SV182: $i,SV213: $i,SV107: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV107 @ SV213 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( SV107 = SV213 )
= $true )
| ( ( ( unordered_pair @ SV107 @ SV182 )
= ( unordered_pair @ SV213 @ SV182 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[894]) ).
thf(931,plain,
! [SV251: $i,SV241: $i,SV183: $i,SV110: $i] :
( ( ( member @ ( ordered_pair @ SV110 @ SV183 ) @ ( cross_product @ SV241 @ SV251 ) )
= $false )
| ( ( ( unordered_pair @ SV110 @ SV183 )
!= null_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[897]) ).
thf(932,plain,
! [SV111: $i,SV184: $i,SV252: $i] :
( ( ( ~ ( member @ SV252 @ SV184 ) )
= $true )
| ( ( subclass @ ( unordered_pair @ SV111 @ SV252 ) @ SV184 )
= $true )
| ( ( member @ SV111 @ SV184 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[898]) ).
thf(933,plain,
! [SV187: $i,SV118: $i] :
( ( ( member @ SV118 @ universal_class )
= $false )
| ( ( SV118 = SV187 )
= $true )
| ( ( ( singleton @ SV118 )
= ( singleton @ SV187 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[899]) ).
thf(934,plain,
! [SV119: $i,SV188: $i] :
( ( ( member @ SV188 @ universal_class )
= $false )
| ( ( SV119 = SV188 )
= $true )
| ( ( ( singleton @ SV119 )
= ( singleton @ SV188 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[900]) ).
thf(935,plain,
! [SV189: $i,SV120: $i,SV214: $i] :
( ( ( ~ ( member @ SV214 @ universal_class ) )
= $true )
| ( ( ( SV214 = SV120 )
| ( SV214 = SV189 ) )
= $true )
| ( ( ( unordered_pair @ SV120 @ SV189 )
= ( singleton @ SV214 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[901]) ).
thf(936,plain,
! [SV125: $i,SV1: $i,SV242: $i] :
( ( ( member @ SV242 @ SV1 )
= $false )
| ( ( member @ SV242 @ SV125 )
= $true )
| ( ( subclass @ SV1 @ SV125 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[902]) ).
thf(937,plain,
! [SV136: $i,SV16: $i,SV253: $i,SV244: $i] :
( ( ( ~ ( member @ SV244 @ SV253 ) )
= $true )
| ( ( member @ ( ordered_pair @ SV16 @ SV244 ) @ ( cross_product @ SV136 @ SV253 ) )
= $true )
| ( ( member @ SV16 @ SV136 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[907]) ).
thf(938,plain,
! [SV141: $i,SV245: $i,SV22: $i] :
( ( ( member @ SV22 @ SV245 )
= $false )
| ( ( member @ SV22 @ ( intersection @ SV141 @ SV245 ) )
= $true )
| ( ( member @ SV22 @ SV141 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[909]) ).
thf(939,plain,
! [SV223: $i,SV149: $i,SV33: $i,SV198: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( rotate @ SV223 ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV149 ) @ SV198 ) @ SV223 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[911]) ).
thf(940,plain,
! [SV225: $i,SV200: $i,SV36: $i,SV151: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( flip @ SV225 ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV36 @ SV151 ) @ SV200 ) @ SV225 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[913]) ).
thf(941,plain,
! [SV204: $i,SV159: $i,SV55: $i,SV227: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( compose @ SV159 @ SV204 ) )
= $true )
| ( ( member @ SV55 @ ( image @ SV159 @ ( image @ SV204 @ ( singleton @ SV227 ) ) ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[916]) ).
thf(942,plain,
! [SV74: $i] :
( ( ( subclass @ ( range_of @ SV74 ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
= $false )
| ( ( operation @ SV74 )
= $true )
| ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV74 ) ) @ ( domain_of @ ( domain_of @ SV74 ) ) )
= ( domain_of @ SV74 ) )
= $false )
| ( ( function @ SV74 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[918]) ).
thf(943,plain,
! [SV231: $i,SV255: $i,SV78: $i] :
( ( ( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV255 ) ) )
| ( compatible @ SV78 @ SV231 @ SV255 ) )
= $true )
| ( ( ( domain_of @ ( domain_of @ SV231 ) )
= ( domain_of @ SV78 ) )
= $false )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[919]) ).
thf(944,plain,
! [SV82: $i,SV209: $i,SV166: $i,SV256: $i,SV254: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV254 @ SV256 ) @ ( domain_of @ SV166 ) )
| ( ( apply @ SV209 @ ( ordered_pair @ ( apply @ SV82 @ SV254 ) @ ( apply @ SV82 @ SV256 ) ) )
= ( apply @ SV82 @ ( apply @ SV166 @ ( ordered_pair @ SV254 @ SV256 ) ) ) ) )
= $true )
| ( ( homomorphism @ SV82 @ SV166 @ SV209 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[920]) ).
thf(945,plain,
! [SV233: $i,SV83: $i,SV257: $i] :
( ( ( ~ ( compatible @ SV257 @ SV83 @ SV233 )
| ( homomorphism @ SV257 @ SV83 @ SV233 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV257 @ SV83 @ SV233 ) @ ( not_homomorphism2 @ SV257 @ SV83 @ SV233 ) ) @ ( domain_of @ SV83 ) ) )
= $true )
| ( ( operation @ SV233 )
= $false )
| ( ( operation @ SV83 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[921]) ).
thf(946,plain,
! [SV234: $i,SV84: $i,SV258: $i] :
( ( ( ~ ( compatible @ SV258 @ SV84 @ SV234 )
| ( ( apply @ SV234 @ ( ordered_pair @ ( apply @ SV258 @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) ) @ ( apply @ SV258 @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) )
!= ( apply @ SV258 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) ) )
| ( homomorphism @ SV258 @ SV84 @ SV234 ) )
= $true )
| ( ( operation @ SV234 )
= $false )
| ( ( operation @ SV84 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[922]) ).
thf(947,plain,
! [SV90: $i,SV250: $i,SV171: $i] :
( ( ( subclass @ SV171 @ SV250 )
= $false )
| ( ( subclass @ SV90 @ SV250 )
= $true )
| ( ( subclass @ SV90 @ SV171 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[927]) ).
thf(948,plain,
! [SV106: $i,SV212: $i,SV181: $i] :
( ( ( member @ ( ordered_pair @ SV181 @ SV212 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( SV181 = SV212 )
= $true )
| ( ( ( unordered_pair @ SV106 @ SV181 )
= ( unordered_pair @ SV106 @ SV212 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[929]) ).
thf(949,plain,
! [SV182: $i,SV213: $i,SV107: $i] :
( ( ( member @ ( ordered_pair @ SV107 @ SV213 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( SV107 = SV213 )
= $true )
| ( ( ( unordered_pair @ SV107 @ SV182 )
= ( unordered_pair @ SV213 @ SV182 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[930]) ).
thf(950,plain,
! [SV251: $i,SV241: $i,SV183: $i,SV110: $i] :
( ( ( ( unordered_pair @ SV110 @ SV183 )
= null_class )
= $false )
| ( ( member @ ( ordered_pair @ SV110 @ SV183 ) @ ( cross_product @ SV241 @ SV251 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[931]) ).
thf(951,plain,
! [SV111: $i,SV184: $i,SV252: $i] :
( ( ( member @ SV252 @ SV184 )
= $false )
| ( ( subclass @ ( unordered_pair @ SV111 @ SV252 ) @ SV184 )
= $true )
| ( ( member @ SV111 @ SV184 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[932]) ).
thf(952,plain,
! [SV189: $i,SV120: $i,SV214: $i] :
( ( ( member @ SV214 @ universal_class )
= $false )
| ( ( ( SV214 = SV120 )
| ( SV214 = SV189 ) )
= $true )
| ( ( ( unordered_pair @ SV120 @ SV189 )
= ( singleton @ SV214 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[935]) ).
thf(953,plain,
! [SV136: $i,SV16: $i,SV253: $i,SV244: $i] :
( ( ( member @ SV244 @ SV253 )
= $false )
| ( ( member @ ( ordered_pair @ SV16 @ SV244 ) @ ( cross_product @ SV136 @ SV253 ) )
= $true )
| ( ( member @ SV16 @ SV136 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[937]) ).
thf(954,plain,
! [SV223: $i,SV149: $i,SV33: $i,SV198: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
= $false )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV198 @ SV33 ) @ SV149 ) @ ( rotate @ SV223 ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV149 ) @ SV198 ) @ SV223 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[939]) ).
thf(955,plain,
! [SV225: $i,SV200: $i,SV36: $i,SV151: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
= $false )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV151 @ SV36 ) @ SV200 ) @ ( flip @ SV225 ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV36 @ SV151 ) @ SV200 ) @ SV225 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[940]) ).
thf(956,plain,
! [SV204: $i,SV159: $i,SV55: $i,SV227: $i] :
( ( ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( member @ ( ordered_pair @ SV227 @ SV55 ) @ ( compose @ SV159 @ SV204 ) )
= $true )
| ( ( member @ SV55 @ ( image @ SV159 @ ( image @ SV204 @ ( singleton @ SV227 ) ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[941]) ).
thf(957,plain,
! [SV231: $i,SV255: $i,SV78: $i] :
( ( ( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV255 ) ) ) )
= $true )
| ( ( compatible @ SV78 @ SV231 @ SV255 )
= $true )
| ( ( ( domain_of @ ( domain_of @ SV231 ) )
= ( domain_of @ SV78 ) )
= $false )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[943]) ).
thf(958,plain,
! [SV82: $i,SV209: $i,SV166: $i,SV256: $i,SV254: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV254 @ SV256 ) @ ( domain_of @ SV166 ) ) )
= $true )
| ( ( ( apply @ SV209 @ ( ordered_pair @ ( apply @ SV82 @ SV254 ) @ ( apply @ SV82 @ SV256 ) ) )
= ( apply @ SV82 @ ( apply @ SV166 @ ( ordered_pair @ SV254 @ SV256 ) ) ) )
= $true )
| ( ( homomorphism @ SV82 @ SV166 @ SV209 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[944]) ).
thf(959,plain,
! [SV233: $i,SV83: $i,SV257: $i] :
( ( ( ~ ( compatible @ SV257 @ SV83 @ SV233 ) )
= $true )
| ( ( ( homomorphism @ SV257 @ SV83 @ SV233 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV257 @ SV83 @ SV233 ) @ ( not_homomorphism2 @ SV257 @ SV83 @ SV233 ) ) @ ( domain_of @ SV83 ) ) )
= $true )
| ( ( operation @ SV233 )
= $false )
| ( ( operation @ SV83 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[945]) ).
thf(960,plain,
! [SV234: $i,SV84: $i,SV258: $i] :
( ( ( ~ ( compatible @ SV258 @ SV84 @ SV234 ) )
= $true )
| ( ( ( ( apply @ SV234 @ ( ordered_pair @ ( apply @ SV258 @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) ) @ ( apply @ SV258 @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) )
!= ( apply @ SV258 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) ) )
| ( homomorphism @ SV258 @ SV84 @ SV234 ) )
= $true )
| ( ( operation @ SV234 )
= $false )
| ( ( operation @ SV84 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[946]) ).
thf(961,plain,
! [SV189: $i,SV120: $i,SV214: $i] :
( ( ( SV214 = SV120 )
= $true )
| ( ( SV214 = SV189 )
= $true )
| ( ( member @ SV214 @ universal_class )
= $false )
| ( ( ( unordered_pair @ SV120 @ SV189 )
= ( singleton @ SV214 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[952]) ).
thf(962,plain,
! [SV231: $i,SV255: $i,SV78: $i] :
( ( ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV255 ) ) )
= $false )
| ( ( compatible @ SV78 @ SV231 @ SV255 )
= $true )
| ( ( ( domain_of @ ( domain_of @ SV231 ) )
= ( domain_of @ SV78 ) )
= $false )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[957]) ).
thf(963,plain,
! [SV82: $i,SV209: $i,SV166: $i,SV256: $i,SV254: $i] :
( ( ( member @ ( ordered_pair @ SV254 @ SV256 ) @ ( domain_of @ SV166 ) )
= $false )
| ( ( ( apply @ SV209 @ ( ordered_pair @ ( apply @ SV82 @ SV254 ) @ ( apply @ SV82 @ SV256 ) ) )
= ( apply @ SV82 @ ( apply @ SV166 @ ( ordered_pair @ SV254 @ SV256 ) ) ) )
= $true )
| ( ( homomorphism @ SV82 @ SV166 @ SV209 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[958]) ).
thf(964,plain,
! [SV233: $i,SV83: $i,SV257: $i] :
( ( ( compatible @ SV257 @ SV83 @ SV233 )
= $false )
| ( ( ( homomorphism @ SV257 @ SV83 @ SV233 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV257 @ SV83 @ SV233 ) @ ( not_homomorphism2 @ SV257 @ SV83 @ SV233 ) ) @ ( domain_of @ SV83 ) ) )
= $true )
| ( ( operation @ SV233 )
= $false )
| ( ( operation @ SV83 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[959]) ).
thf(965,plain,
! [SV234: $i,SV84: $i,SV258: $i] :
( ( ( compatible @ SV258 @ SV84 @ SV234 )
= $false )
| ( ( ( ( apply @ SV234 @ ( ordered_pair @ ( apply @ SV258 @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) ) @ ( apply @ SV258 @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) )
!= ( apply @ SV258 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) ) )
| ( homomorphism @ SV258 @ SV84 @ SV234 ) )
= $true )
| ( ( operation @ SV234 )
= $false )
| ( ( operation @ SV84 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[960]) ).
thf(966,plain,
! [SV233: $i,SV83: $i,SV257: $i] :
( ( ( homomorphism @ SV257 @ SV83 @ SV233 )
= $true )
| ( ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV257 @ SV83 @ SV233 ) @ ( not_homomorphism2 @ SV257 @ SV83 @ SV233 ) ) @ ( domain_of @ SV83 ) )
= $true )
| ( ( compatible @ SV257 @ SV83 @ SV233 )
= $false )
| ( ( operation @ SV233 )
= $false )
| ( ( operation @ SV83 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[964]) ).
thf(967,plain,
! [SV84: $i,SV258: $i,SV234: $i] :
( ( ( ( apply @ SV234 @ ( ordered_pair @ ( apply @ SV258 @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) ) @ ( apply @ SV258 @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) )
!= ( apply @ SV258 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) ) )
= $true )
| ( ( homomorphism @ SV258 @ SV84 @ SV234 )
= $true )
| ( ( compatible @ SV258 @ SV84 @ SV234 )
= $false )
| ( ( operation @ SV234 )
= $false )
| ( ( operation @ SV84 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[965]) ).
thf(968,plain,
! [SV84: $i,SV258: $i,SV234: $i] :
( ( ( ( apply @ SV234 @ ( ordered_pair @ ( apply @ SV258 @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) ) @ ( apply @ SV258 @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) )
= ( apply @ SV258 @ ( apply @ SV84 @ ( ordered_pair @ ( not_homomorphism1 @ SV258 @ SV84 @ SV234 ) @ ( not_homomorphism2 @ SV258 @ SV84 @ SV234 ) ) ) ) )
= $false )
| ( ( homomorphism @ SV258 @ SV84 @ SV234 )
= $true )
| ( ( compatible @ SV258 @ SV84 @ SV234 )
= $false )
| ( ( operation @ SV234 )
= $false )
| ( ( operation @ SV84 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[967]) ).
thf(969,plain,
$false = $true,
inference(fo_atp_e,[status(thm)],[318,968,966,963,962,961,956,955,954,953,951,950,949,948,947,942,938,936,934,933,928,926,925,924,923,917,915,914,912,910,908,906,905,904,903,896,895,891,890,881,880,879,877,876,875,873,871,868,862,861,860,859,857,856,854,844,843,836,832,814,807,794,789,788,785,784,783,781,780,774,766,765,764,763,742,741,740,738,737,733,732,731,730,727,726,725,723,722,719,718,711,710,691,689,688,682,678,668,667,666,665,663,634,633,632,623,614,603,602,590,589,579,566,551,543,524,523,505,503,496,492,491,488,485,466,458,454,443,426,394,393,388,371,369,363,337]) ).
thf(970,plain,
$false,
inference(solved_all_splits,[solved_all_splits(join,[])],[969]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET090-7 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.03 % Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.07/0.20 % Computer : n017.cluster.edu
% 0.07/0.20 % Model : x86_64 x86_64
% 0.07/0.20 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.20 % Memory : 8046.5625MB
% 0.07/0.20 % OS : Linux 6.8.0-71-generic
% 0.07/0.20 % CPULimit : 300
% 0.07/0.20 % WCLimit : 300
% 0.07/0.20 % DateTime : Tue Oct 6 14:26:16 UTC 2026
% 0.07/0.20 % CPUTime :
% 0.07/0.20 Running leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 7.73/1.53 .
% 7.73/1.53
% 7.73/1.53 No.of.Axioms: 135
% 7.73/1.53
% 7.73/1.53 Length.of.Defs: 0
% 7.73/1.53
% 7.73/1.53 Contains.Choice.Funs: false
% 7.73/1.53 ..............................................
% 7.73/1.53
% 7.73/1.53 ********************************
% 7.73/1.53 * All subproblems solved! *
% 7.73/1.53 ********************************
% 7.73/1.53 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:2,axioms:135,ps:3,u:5,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:true,expand_extuni:false,foatp:e,atp_timeout:74,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:969,loop_count:0,foatp_calls:1,translation:fof_experiment)
% 7.73/1.53
% 7.73/1.53 %**** Beginning of derivation protocol ****
% 7.73/1.53 % SZS output start CNFRefutation
% See solution above
% 7.73/1.54
% 7.73/1.54 %**** End of derivation protocol ****
% 7.73/1.54 %**** no. of clauses in derivation: 970 ****
% 7.73/1.54 %**** clause counter: 969 ****
% 7.73/1.54
% 7.73/1.54 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:2,axioms:135,ps:3,u:5,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:true,expand_extuni:false,foatp:e,atp_timeout:74,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:969,loop_count:0,foatp_calls:1,translation:fof_experiment)
%------------------------------------------------------------------------------