%------------------------------------------------------------------------------
% File : LEO-II---2.3.1
% Problem : SET080-6 : 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 : n016.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 10:39:25 AM UTC 2026
% Result : Unsatisfiable 1.74s 0.62s
% Output : CNFRefutation 1.74s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 93
% Syntax : Number of formulae : 642 ( 406 unt; 0 typ; 0 def)
% Number of atoms : 3023 (1046 equ; 0 cnn)
% Maximal formula atoms : 5 ( 4 avg)
% Number of connectives : 6877 ( 617 ~; 807 |; 0 &;5453 @)
% ( 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 : 50 ( 47 usr; 10 con; 0-3 aty)
% Number of variables : 1418 ( 0 ^;1418 !; 0 ?;1418 :)
% 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_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_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,
! [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(2,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(3,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(4,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(5,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(6,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf1 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(7,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(8,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(9,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(10,axiom,
! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( function @ Xh ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(11,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(12,axiom,
! [Xf: $i] :
( ~ ( operation @ Xf )
| ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(13,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(14,axiom,
! [Xf: $i] :
( ~ ( operation @ Xf )
| ( function @ Xf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(15,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(16,axiom,
! [Xr: $i] :
( ( complement @ ( domain_of @ ( intersection @ Xr @ identity_relation ) ) )
= ( diagonalise @ Xr ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(17,axiom,
( ( intersection @ ( inverse @ subset_relation ) @ subset_relation )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(18,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(19,axiom,
! [Xf: $i] :
( ~ ( function @ ( inverse @ Xf ) )
| ~ ( function @ Xf )
| ( one_to_one @ Xf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one3) ).
thf(20,axiom,
! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ ( inverse @ Xf ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one2) ).
thf(21,axiom,
! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ Xf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one1) ).
thf(22,axiom,
! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( Y = null_class )
| ( member @ ( apply @ choice @ Y ) @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(23,axiom,
function @ choice,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(24,axiom,
! [Xf: $i,Y: $i] :
( ( sum_class @ ( image @ Xf @ ( singleton @ Y ) ) )
= ( apply @ Xf @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(25,axiom,
! [X: $i] :
( ( X = null_class )
| ( ( intersection @ X @ ( regular @ X ) )
= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(26,axiom,
! [X: $i] :
( ( X = null_class )
| ( member @ ( regular @ X ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(27,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(28,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(29,axiom,
! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(30,axiom,
! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(31,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(32,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(33,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(34,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(35,axiom,
! [Yr: $i,Xr: $i] : ( subclass @ ( compose @ Yr @ Xr ) @ ( cross_product @ universal_class @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(36,axiom,
! [U: $i] :
( ~ ( member @ U @ universal_class )
| ( member @ ( power_class @ U ) @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(37,axiom,
! [X: $i] :
( ( complement @ ( image @ element_relation @ ( complement @ X ) ) )
= ( power_class @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(38,axiom,
! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ ( sum_class @ X ) @ universal_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(39,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(40,axiom,
member @ omega @ universal_class,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(41,axiom,
! [Y: $i] :
( ~ ( inductive @ Y )
| ( subclass @ omega @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(42,axiom,
inductive @ omega,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(43,axiom,
! [X: $i] :
( ~ ( member @ null_class @ X )
| ~ ( subclass @ ( image @ successor_relation @ X ) @ X )
| ( inductive @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(44,axiom,
! [X: $i] :
( ~ ( inductive @ X )
| ( subclass @ ( image @ successor_relation @ X ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(45,axiom,
! [X: $i] :
( ~ ( inductive @ X )
| ( member @ null_class @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(46,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(47,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(48,axiom,
subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(49,axiom,
! [X: $i] :
( ( union @ X @ ( singleton @ X ) )
= ( successor @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(50,axiom,
! [Xr: $i,X: $i] :
( ( range_of @ ( restrict @ Xr @ X @ universal_class ) )
= ( image @ Xr @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(51,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(52,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(53,axiom,
! [Z: $i] :
( ( domain_of @ ( inverse @ Z ) )
= ( range_of @ Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(54,axiom,
! [Y: $i] :
( ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) )
= ( inverse @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(55,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(56,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(57,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(58,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(59,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(60,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(61,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(62,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(63,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(64,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(65,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(66,axiom,
! [X: $i,Y: $i] :
( ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) )
= ( union @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(67,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(68,axiom,
! [Z: $i,X: $i] :
( ~ ( member @ Z @ ( complement @ X ) )
| ~ ( member @ Z @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(69,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(70,axiom,
! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(71,axiom,
! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(72,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(73,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(74,axiom,
subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(75,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(76,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(77,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(78,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(79,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(80,axiom,
! [X: $i] :
( ( unordered_pair @ X @ X )
= ( singleton @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(81,axiom,
! [X: $i,Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(82,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(83,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(84,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(85,axiom,
! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( subclass @ Y @ X )
| ( X = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(86,axiom,
! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ Y @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(87,axiom,
! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ X @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(88,axiom,
! [X: $i] : ( subclass @ X @ universal_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(89,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(90,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(91,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(92,conjecture,
$false,
file('no conjecture given, we try to refute the axioms',dummy_conjecture) ).
thf(93,negated_conjecture,
$false = $false,
inference(negate_conjecture,[status(cth)],[92]) ).
thf(94,negated_conjecture,
~ ( member @ null_class @ ( singleton @ null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_null_class_in_its_singleton_1) ).
thf(95,plain,
$false = $false,
inference(unfold_def,[status(thm)],[93]) ).
thf(96,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)],[1]) ).
thf(97,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)],[2]) ).
thf(98,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)],[3]) ).
thf(99,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[4]) ).
thf(100,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[5]) ).
thf(101,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[6]) ).
thf(102,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)],[7]) ).
thf(103,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)],[8]) ).
thf(104,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)],[9]) ).
thf(105,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( function @ Xh ) ) )
= $true ),
inference(unfold_def,[status(thm)],[10]) ).
thf(106,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)],[11]) ).
thf(107,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[12]) ).
thf(108,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)],[13]) ).
thf(109,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( function @ Xf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[14]) ).
thf(110,plain,
( ( ! [X: $i] :
( ( intersection @ ( domain_of @ X ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ X ) ) )
= ( cantor @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[15]) ).
thf(111,plain,
( ( ! [Xr: $i] :
( ( complement @ ( domain_of @ ( intersection @ Xr @ identity_relation ) ) )
= ( diagonalise @ Xr ) ) )
= $true ),
inference(unfold_def,[status(thm)],[16]) ).
thf(112,plain,
( ( ( intersection @ ( inverse @ subset_relation ) @ subset_relation )
= identity_relation )
= $true ),
inference(unfold_def,[status(thm)],[17]) ).
thf(113,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)],[18]) ).
thf(114,plain,
( ( ! [Xf: $i] :
( ~ ( function @ ( inverse @ Xf ) )
| ~ ( function @ Xf )
| ( one_to_one @ Xf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[19]) ).
thf(115,plain,
( ( ! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ ( inverse @ Xf ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[20]) ).
thf(116,plain,
( ( ! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ Xf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[21]) ).
thf(117,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( Y = null_class )
| ( member @ ( apply @ choice @ Y ) @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[22]) ).
thf(118,plain,
( ( function @ choice )
= $true ),
inference(unfold_def,[status(thm)],[23]) ).
thf(119,plain,
( ( ! [Xf: $i,Y: $i] :
( ( sum_class @ ( image @ Xf @ ( singleton @ Y ) ) )
= ( apply @ Xf @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[24]) ).
thf(120,plain,
( ( ! [X: $i] :
( ( X = null_class )
| ( ( intersection @ X @ ( regular @ X ) )
= null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[25]) ).
thf(121,plain,
( ( ! [X: $i] :
( ( X = null_class )
| ( member @ ( regular @ X ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[26]) ).
thf(122,plain,
( ( ! [Xf: $i,X: $i] :
( ~ ( function @ Xf )
| ~ ( member @ X @ universal_class )
| ( member @ ( image @ Xf @ X ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[27]) ).
thf(123,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)],[28]) ).
thf(124,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation ) ) )
= $true ),
inference(unfold_def,[status(thm)],[29]) ).
thf(125,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[30]) ).
thf(126,plain,
( ( ! [X: $i] :
( ~ ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation )
| ( single_valued_class @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[31]) ).
thf(127,plain,
( ( ! [X: $i] :
( ~ ( single_valued_class @ X )
| ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation ) ) )
= $true ),
inference(unfold_def,[status(thm)],[32]) ).
thf(128,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)],[33]) ).
thf(129,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)],[34]) ).
thf(130,plain,
( ( ! [Yr: $i,Xr: $i] : ( subclass @ ( compose @ Yr @ Xr ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[35]) ).
thf(131,plain,
( ( ! [U: $i] :
( ~ ( member @ U @ universal_class )
| ( member @ ( power_class @ U ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[36]) ).
thf(132,plain,
( ( ! [X: $i] :
( ( complement @ ( image @ element_relation @ ( complement @ X ) ) )
= ( power_class @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[37]) ).
thf(133,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ ( sum_class @ X ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[38]) ).
thf(134,plain,
( ( ! [X: $i] :
( ( domain_of @ ( restrict @ element_relation @ universal_class @ X ) )
= ( sum_class @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[39]) ).
thf(135,plain,
( ( member @ omega @ universal_class )
= $true ),
inference(unfold_def,[status(thm)],[40]) ).
thf(136,plain,
( ( ! [Y: $i] :
( ~ ( inductive @ Y )
| ( subclass @ omega @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[41]) ).
thf(137,plain,
( ( inductive @ omega )
= $true ),
inference(unfold_def,[status(thm)],[42]) ).
thf(138,plain,
( ( ! [X: $i] :
( ~ ( member @ null_class @ X )
| ~ ( subclass @ ( image @ successor_relation @ X ) @ X )
| ( inductive @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[43]) ).
thf(139,plain,
( ( ! [X: $i] :
( ~ ( inductive @ X )
| ( subclass @ ( image @ successor_relation @ X ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[44]) ).
thf(140,plain,
( ( ! [X: $i] :
( ~ ( inductive @ X )
| ( member @ null_class @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[45]) ).
thf(141,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)],[46]) ).
thf(142,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ successor_relation )
| ( ( successor @ X )
= Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[47]) ).
thf(143,plain,
( ( subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[48]) ).
thf(144,plain,
( ( ! [X: $i] :
( ( union @ X @ ( singleton @ X ) )
= ( successor @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[49]) ).
thf(145,plain,
( ( ! [Xr: $i,X: $i] :
( ( range_of @ ( restrict @ Xr @ X @ universal_class ) )
= ( image @ Xr @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[50]) ).
thf(146,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)],[51]) ).
thf(147,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)],[52]) ).
thf(148,plain,
( ( ! [Z: $i] :
( ( domain_of @ ( inverse @ Z ) )
= ( range_of @ Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[53]) ).
thf(149,plain,
( ( ! [Y: $i] :
( ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) )
= ( inverse @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[54]) ).
thf(150,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)],[55]) ).
thf(151,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)],[56]) ).
thf(152,plain,
( ( ! [X: $i] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[57]) ).
thf(153,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)],[58]) ).
thf(154,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)],[59]) ).
thf(155,plain,
( ( ! [X: $i] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[60]) ).
thf(156,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)],[61]) ).
thf(157,plain,
( ( ! [X: $i,Z: $i] :
( ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
!= null_class )
| ~ ( member @ Z @ ( domain_of @ X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[62]) ).
thf(158,plain,
( ( ! [X: $i,Y: $i,Xr: $i] :
( ( intersection @ ( cross_product @ X @ Y ) @ Xr )
= ( restrict @ Xr @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[63]) ).
thf(159,plain,
( ( ! [Xr: $i,X: $i,Y: $i] :
( ( intersection @ Xr @ ( cross_product @ X @ Y ) )
= ( restrict @ Xr @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[64]) ).
thf(160,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)],[65]) ).
thf(161,plain,
( ( ! [X: $i,Y: $i] :
( ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) )
= ( union @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[66]) ).
thf(162,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ universal_class )
| ( member @ Z @ ( complement @ X ) )
| ( member @ Z @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[67]) ).
thf(163,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ ( complement @ X ) )
| ~ ( member @ Z @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[68]) ).
thf(164,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ X )
| ~ ( member @ Z @ Y )
| ( member @ Z @ ( intersection @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[69]) ).
thf(165,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[70]) ).
thf(166,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[71]) ).
thf(167,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)],[72]) ).
thf(168,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ element_relation )
| ( member @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[73]) ).
thf(169,plain,
( ( subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[74]) ).
thf(170,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)],[75]) ).
thf(171,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)],[76]) ).
thf(172,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)],[77]) ).
thf(173,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)],[78]) ).
thf(174,plain,
( ( ! [X: $i,Y: $i] :
( ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) )
= ( ordered_pair @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[79]) ).
thf(175,plain,
( ( ! [X: $i] :
( ( unordered_pair @ X @ X )
= ( singleton @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[80]) ).
thf(176,plain,
( ( ! [X: $i,Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[81]) ).
thf(177,plain,
( ( ! [Y: $i,X: $i] :
( ~ ( member @ Y @ universal_class )
| ( member @ Y @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[82]) ).
thf(178,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ X @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[83]) ).
thf(179,plain,
( ( ! [U: $i,X: $i,Y: $i] :
( ~ ( member @ U @ ( unordered_pair @ X @ Y ) )
| ( U = X )
| ( U = Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[84]) ).
thf(180,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( subclass @ Y @ X )
| ( X = Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[85]) ).
thf(181,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ Y @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[86]) ).
thf(182,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[87]) ).
thf(183,plain,
( ( ! [X: $i] : ( subclass @ X @ universal_class ) )
= $true ),
inference(unfold_def,[status(thm)],[88]) ).
thf(184,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[89]) ).
thf(185,plain,
( ( ! [X: $i,Y: $i] :
( ( member @ ( not_subclass_element @ X @ Y ) @ X )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[90]) ).
thf(186,plain,
( ( ! [X: $i,Y: $i,U: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( member @ U @ X )
| ( member @ U @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[91]) ).
thf(187,plain,
( ( ~ ( member @ null_class @ ( singleton @ null_class ) ) )
= $true ),
inference(unfold_def,[status(thm)],[94]) ).
thf(188,plain,
( ( ~ $false ) = $true ),
inference(polarity_switch,[status(thm)],[95]) ).
thf(189,plain,
( ( ~ ( member @ null_class @ ( singleton @ null_class ) ) )
= $true ),
inference(copy,[status(thm)],[187]) ).
thf(190,plain,
( ( ! [X: $i,Y: $i,U: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( member @ U @ X )
| ( member @ U @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[186]) ).
thf(191,plain,
( ( ! [X: $i,Y: $i] :
( ( member @ ( not_subclass_element @ X @ Y ) @ X )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[185]) ).
thf(192,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( not_subclass_element @ X @ Y ) @ Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[184]) ).
thf(193,plain,
( ( ! [X: $i] : ( subclass @ X @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[183]) ).
thf(194,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[182]) ).
thf(195,plain,
( ( ! [X: $i,Y: $i] :
( ( X != Y )
| ( subclass @ Y @ X ) ) )
= $true ),
inference(copy,[status(thm)],[181]) ).
thf(196,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( subclass @ X @ Y )
| ~ ( subclass @ Y @ X )
| ( X = Y ) ) )
= $true ),
inference(copy,[status(thm)],[180]) ).
thf(197,plain,
( ( ! [U: $i,X: $i,Y: $i] :
( ~ ( member @ U @ ( unordered_pair @ X @ Y ) )
| ( U = X )
| ( U = Y ) ) )
= $true ),
inference(copy,[status(thm)],[179]) ).
thf(198,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ X @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[178]) ).
thf(199,plain,
( ( ! [Y: $i,X: $i] :
( ~ ( member @ Y @ universal_class )
| ( member @ Y @ ( unordered_pair @ X @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[177]) ).
thf(200,plain,
( ( ! [X: $i,Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[176]) ).
thf(201,plain,
( ( ! [X: $i] :
( ( unordered_pair @ X @ X )
= ( singleton @ X ) ) )
= $true ),
inference(copy,[status(thm)],[175]) ).
thf(202,plain,
( ( ! [X: $i,Y: $i] :
( ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) )
= ( ordered_pair @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[174]) ).
thf(203,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)],[173]) ).
thf(204,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)],[172]) ).
thf(205,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(copy,[status(thm)],[171]) ).
thf(206,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)],[170]) ).
thf(207,plain,
( ( subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[169]) ).
thf(208,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ element_relation )
| ( member @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[168]) ).
thf(209,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)],[167]) ).
thf(210,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ X ) ) )
= $true ),
inference(copy,[status(thm)],[166]) ).
thf(211,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ ( intersection @ X @ Y ) )
| ( member @ Z @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[165]) ).
thf(212,plain,
( ( ! [Z: $i,X: $i,Y: $i] :
( ~ ( member @ Z @ X )
| ~ ( member @ Z @ Y )
| ( member @ Z @ ( intersection @ X @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[164]) ).
thf(213,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ ( complement @ X ) )
| ~ ( member @ Z @ X ) ) )
= $true ),
inference(copy,[status(thm)],[163]) ).
thf(214,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ universal_class )
| ( member @ Z @ ( complement @ X ) )
| ( member @ Z @ X ) ) )
= $true ),
inference(copy,[status(thm)],[162]) ).
thf(215,plain,
( ( ! [X: $i,Y: $i] :
( ( complement @ ( intersection @ ( complement @ X ) @ ( complement @ Y ) ) )
= ( union @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[161]) ).
thf(216,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)],[160]) ).
thf(217,plain,
( ( ! [Xr: $i,X: $i,Y: $i] :
( ( intersection @ Xr @ ( cross_product @ X @ Y ) )
= ( restrict @ Xr @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[159]) ).
thf(218,plain,
( ( ! [X: $i,Y: $i,Xr: $i] :
( ( intersection @ ( cross_product @ X @ Y ) @ Xr )
= ( restrict @ Xr @ X @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[158]) ).
thf(219,plain,
( ( ! [X: $i,Z: $i] :
( ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
!= null_class )
| ~ ( member @ Z @ ( domain_of @ X ) ) ) )
= $true ),
inference(copy,[status(thm)],[157]) ).
thf(220,plain,
( ( ! [Z: $i,X: $i] :
( ~ ( member @ Z @ universal_class )
| ( ( restrict @ X @ ( singleton @ Z ) @ universal_class )
= null_class )
| ( member @ Z @ ( domain_of @ X ) ) ) )
= $true ),
inference(copy,[status(thm)],[156]) ).
thf(221,plain,
( ( ! [X: $i] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[155]) ).
thf(222,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)],[154]) ).
thf(223,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)],[153]) ).
thf(224,plain,
( ( ! [X: $i] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[152]) ).
thf(225,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)],[151]) ).
thf(226,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)],[150]) ).
thf(227,plain,
( ( ! [Y: $i] :
( ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) )
= ( inverse @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[149]) ).
thf(228,plain,
( ( ! [Z: $i] :
( ( domain_of @ ( inverse @ Z ) )
= ( range_of @ Z ) ) )
= $true ),
inference(copy,[status(thm)],[148]) ).
thf(229,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)],[147]) ).
thf(230,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)],[146]) ).
thf(231,plain,
( ( ! [Xr: $i,X: $i] :
( ( range_of @ ( restrict @ Xr @ X @ universal_class ) )
= ( image @ Xr @ X ) ) )
= $true ),
inference(copy,[status(thm)],[145]) ).
thf(232,plain,
( ( ! [X: $i] :
( ( union @ X @ ( singleton @ X ) )
= ( successor @ X ) ) )
= $true ),
inference(copy,[status(thm)],[144]) ).
thf(233,plain,
( ( subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(copy,[status(thm)],[143]) ).
thf(234,plain,
( ( ! [X: $i,Y: $i] :
( ~ ( member @ ( ordered_pair @ X @ Y ) @ successor_relation )
| ( ( successor @ X )
= Y ) ) )
= $true ),
inference(copy,[status(thm)],[142]) ).
thf(235,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)],[141]) ).
thf(236,plain,
( ( ! [X: $i] :
( ~ ( inductive @ X )
| ( member @ null_class @ X ) ) )
= $true ),
inference(copy,[status(thm)],[140]) ).
thf(237,plain,
( ( ! [X: $i] :
( ~ ( inductive @ X )
| ( subclass @ ( image @ successor_relation @ X ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[139]) ).
thf(238,plain,
( ( ! [X: $i] :
( ~ ( member @ null_class @ X )
| ~ ( subclass @ ( image @ successor_relation @ X ) @ X )
| ( inductive @ X ) ) )
= $true ),
inference(copy,[status(thm)],[138]) ).
thf(239,plain,
( ( inductive @ omega )
= $true ),
inference(copy,[status(thm)],[137]) ).
thf(240,plain,
( ( ! [Y: $i] :
( ~ ( inductive @ Y )
| ( subclass @ omega @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[136]) ).
thf(241,plain,
( ( member @ omega @ universal_class )
= $true ),
inference(copy,[status(thm)],[135]) ).
thf(242,plain,
( ( ! [X: $i] :
( ( domain_of @ ( restrict @ element_relation @ universal_class @ X ) )
= ( sum_class @ X ) ) )
= $true ),
inference(copy,[status(thm)],[134]) ).
thf(243,plain,
( ( ! [X: $i] :
( ~ ( member @ X @ universal_class )
| ( member @ ( sum_class @ X ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[133]) ).
thf(244,plain,
( ( ! [X: $i] :
( ( complement @ ( image @ element_relation @ ( complement @ X ) ) )
= ( power_class @ X ) ) )
= $true ),
inference(copy,[status(thm)],[132]) ).
thf(245,plain,
( ( ! [U: $i] :
( ~ ( member @ U @ universal_class )
| ( member @ ( power_class @ U ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[131]) ).
thf(246,plain,
( ( ! [Yr: $i,Xr: $i] : ( subclass @ ( compose @ Yr @ Xr ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[130]) ).
thf(247,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)],[129]) ).
thf(248,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)],[128]) ).
thf(249,plain,
( ( ! [X: $i] :
( ~ ( single_valued_class @ X )
| ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation ) ) )
= $true ),
inference(copy,[status(thm)],[127]) ).
thf(250,plain,
( ( ! [X: $i] :
( ~ ( subclass @ ( compose @ X @ ( inverse @ X ) ) @ identity_relation )
| ( single_valued_class @ X ) ) )
= $true ),
inference(copy,[status(thm)],[126]) ).
thf(251,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ Xf @ ( cross_product @ universal_class @ universal_class ) ) ) )
= $true ),
inference(copy,[status(thm)],[125]) ).
thf(252,plain,
( ( ! [Xf: $i] :
( ~ ( function @ Xf )
| ( subclass @ ( compose @ Xf @ ( inverse @ Xf ) ) @ identity_relation ) ) )
= $true ),
inference(copy,[status(thm)],[124]) ).
thf(253,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)],[123]) ).
thf(254,plain,
( ( ! [Xf: $i,X: $i] :
( ~ ( function @ Xf )
| ~ ( member @ X @ universal_class )
| ( member @ ( image @ Xf @ X ) @ universal_class ) ) )
= $true ),
inference(copy,[status(thm)],[122]) ).
thf(255,plain,
( ( ! [X: $i] :
( ( X = null_class )
| ( member @ ( regular @ X ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[121]) ).
thf(256,plain,
( ( ! [X: $i] :
( ( X = null_class )
| ( ( intersection @ X @ ( regular @ X ) )
= null_class ) ) )
= $true ),
inference(copy,[status(thm)],[120]) ).
thf(257,plain,
( ( ! [Xf: $i,Y: $i] :
( ( sum_class @ ( image @ Xf @ ( singleton @ Y ) ) )
= ( apply @ Xf @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[119]) ).
thf(258,plain,
( ( function @ choice )
= $true ),
inference(copy,[status(thm)],[118]) ).
thf(259,plain,
( ( ! [Y: $i] :
( ~ ( member @ Y @ universal_class )
| ( Y = null_class )
| ( member @ ( apply @ choice @ Y ) @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[117]) ).
thf(260,plain,
( ( ! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ Xf ) ) )
= $true ),
inference(copy,[status(thm)],[116]) ).
thf(261,plain,
( ( ! [Xf: $i] :
( ~ ( one_to_one @ Xf )
| ( function @ ( inverse @ Xf ) ) ) )
= $true ),
inference(copy,[status(thm)],[115]) ).
thf(262,plain,
( ( ! [Xf: $i] :
( ~ ( function @ ( inverse @ Xf ) )
| ~ ( function @ Xf )
| ( one_to_one @ Xf ) ) )
= $true ),
inference(copy,[status(thm)],[114]) ).
thf(263,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)],[113]) ).
thf(264,plain,
( ( ( intersection @ ( inverse @ subset_relation ) @ subset_relation )
= identity_relation )
= $true ),
inference(copy,[status(thm)],[112]) ).
thf(265,plain,
( ( ! [Xr: $i] :
( ( complement @ ( domain_of @ ( intersection @ Xr @ identity_relation ) ) )
= ( diagonalise @ Xr ) ) )
= $true ),
inference(copy,[status(thm)],[111]) ).
thf(266,plain,
( ( ! [X: $i] :
( ( intersection @ ( domain_of @ X ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ X ) ) )
= ( cantor @ X ) ) )
= $true ),
inference(copy,[status(thm)],[110]) ).
thf(267,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( function @ Xf ) ) )
= $true ),
inference(copy,[status(thm)],[109]) ).
thf(268,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)],[108]) ).
thf(269,plain,
( ( ! [Xf: $i] :
( ~ ( operation @ Xf )
| ( subclass @ ( range_of @ Xf ) @ ( domain_of @ ( domain_of @ Xf ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[107]) ).
thf(270,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)],[106]) ).
thf(271,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( function @ Xh ) ) )
= $true ),
inference(copy,[status(thm)],[105]) ).
thf(272,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( compatible @ Xh @ Xf1 @ Xf2 )
| ( ( domain_of @ ( domain_of @ Xf1 ) )
= ( domain_of @ Xh ) ) ) )
= $true ),
inference(copy,[status(thm)],[104]) ).
thf(273,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)],[103]) ).
thf(274,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(copy,[status(thm)],[102]) ).
thf(275,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf1 ) ) )
= $true ),
inference(copy,[status(thm)],[101]) ).
thf(276,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( operation @ Xf2 ) ) )
= $true ),
inference(copy,[status(thm)],[100]) ).
thf(277,plain,
( ( ! [Xh: $i,Xf1: $i,Xf2: $i] :
( ~ ( homomorphism @ Xh @ Xf1 @ Xf2 )
| ( compatible @ Xh @ Xf1 @ Xf2 ) ) )
= $true ),
inference(copy,[status(thm)],[99]) ).
thf(278,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(copy,[status(thm)],[98]) ).
thf(279,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(copy,[status(thm)],[97]) ).
thf(280,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(copy,[status(thm)],[96]) ).
thf(281,plain,
( ( ~ $false ) = $true ),
inference(copy,[status(thm)],[188]) ).
thf(282,plain,
( ( ! [SX0: $i,SX1: $i,SX2: $i] :
( ~ ( function @ SX0 )
| ( ( domain_of @ ( domain_of @ SX1 ) )
!= ( domain_of @ SX0 ) )
| ~ ( subclass @ ( range_of @ SX0 ) @ ( domain_of @ ( domain_of @ SX2 ) ) )
| ( compatible @ SX0 @ SX1 @ SX2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[274]) ).
thf(283,plain,
( ( ! [SX0: $i,SX1: $i] :
( ( SX0 != SX1 )
| ( subclass @ SX0 @ SX1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[194]) ).
thf(284,plain,
( ( ! [SX0: $i,SX1: $i] :
( ( SX0 != SX1 )
| ( subclass @ SX1 @ SX0 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[195]) ).
thf(285,plain,
( ( ! [SX0: $i] :
( ~ ( function @ SX0 )
| ( ( cross_product @ ( domain_of @ ( domain_of @ SX0 ) ) @ ( domain_of @ ( domain_of @ SX0 ) ) )
!= ( domain_of @ SX0 ) )
| ~ ( subclass @ ( range_of @ SX0 ) @ ( domain_of @ ( domain_of @ SX0 ) ) )
| ( operation @ SX0 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[270]) ).
thf(286,plain,
( ( ! [SX0: $i,SX1: $i,SX2: $i] :
( ~ ( operation @ SX0 )
| ~ ( operation @ SX1 )
| ~ ( compatible @ SX2 @ SX0 @ SX1 )
| ( ( apply @ SX1 @ ( ordered_pair @ ( apply @ SX2 @ ( not_homomorphism1 @ SX2 @ SX0 @ SX1 ) ) @ ( apply @ SX2 @ ( not_homomorphism2 @ SX2 @ SX0 @ SX1 ) ) ) )
!= ( apply @ SX2 @ ( apply @ SX0 @ ( ordered_pair @ ( not_homomorphism1 @ SX2 @ SX0 @ SX1 ) @ ( not_homomorphism2 @ SX2 @ SX0 @ SX1 ) ) ) ) )
| ( homomorphism @ SX2 @ SX0 @ SX1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[280]) ).
thf(287,plain,
( ( ! [SX0: $i,SX1: $i] :
( ( ( successor @ SX0 )
!= SX1 )
| ~ ( member @ ( ordered_pair @ SX0 @ SX1 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SX0 @ SX1 ) @ successor_relation ) ) )
= $true ),
inference(unfold_def,[status(thm)],[235]) ).
thf(288,plain,
( ( ! [SX0: $i,SX1: $i] :
( ( ( restrict @ SX0 @ ( singleton @ SX1 ) @ universal_class )
!= null_class )
| ~ ( member @ SX1 @ ( domain_of @ SX0 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[219]) ).
thf(289,plain,
( ( member @ null_class @ ( singleton @ null_class ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[189]) ).
thf(290,plain,
! [SV1: $i] :
( ( ! [SY176: $i,SY177: $i] :
( ~ ( subclass @ SV1 @ SY176 )
| ~ ( member @ SY177 @ SV1 )
| ( member @ SY177 @ SY176 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[190]) ).
thf(291,plain,
! [SV2: $i] :
( ( ! [SY178: $i] :
( ( member @ ( not_subclass_element @ SV2 @ SY178 ) @ SV2 )
| ( subclass @ SV2 @ SY178 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[191]) ).
thf(292,plain,
! [SV3: $i] :
( ( ! [SY179: $i] :
( ~ ( member @ ( not_subclass_element @ SV3 @ SY179 ) @ SY179 )
| ( subclass @ SV3 @ SY179 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[192]) ).
thf(293,plain,
! [SV4: $i] :
( ( subclass @ SV4 @ universal_class )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[193]) ).
thf(294,plain,
! [SV5: $i] :
( ( ! [SY180: $i] :
( ~ ( subclass @ SV5 @ SY180 )
| ~ ( subclass @ SY180 @ SV5 )
| ( SV5 = SY180 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[196]) ).
thf(295,plain,
! [SV6: $i] :
( ( ! [SY181: $i,SY182: $i] :
( ~ ( member @ SV6 @ ( unordered_pair @ SY181 @ SY182 ) )
| ( SV6 = SY181 )
| ( SV6 = SY182 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[197]) ).
thf(296,plain,
! [SV7: $i] :
( ( ! [SY183: $i] :
( ~ ( member @ SV7 @ universal_class )
| ( member @ SV7 @ ( unordered_pair @ SV7 @ SY183 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[198]) ).
thf(297,plain,
! [SV8: $i] :
( ( ! [SY184: $i] :
( ~ ( member @ SV8 @ universal_class )
| ( member @ SV8 @ ( unordered_pair @ SY184 @ SV8 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[199]) ).
thf(298,plain,
! [SV9: $i] :
( ( ! [SY185: $i] : ( member @ ( unordered_pair @ SV9 @ SY185 ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[200]) ).
thf(299,plain,
! [SV10: $i] :
( ( ( unordered_pair @ SV10 @ SV10 )
= ( singleton @ SV10 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[201]) ).
thf(300,plain,
! [SV11: $i] :
( ( ! [SY186: $i] :
( ( unordered_pair @ ( singleton @ SV11 ) @ ( unordered_pair @ SV11 @ ( singleton @ SY186 ) ) )
= ( ordered_pair @ SV11 @ SY186 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[202]) ).
thf(301,plain,
! [SV12: $i] :
( ( ! [SY187: $i,SY188: $i,SY189: $i] :
( ~ ( member @ ( ordered_pair @ SV12 @ SY187 ) @ ( cross_product @ SY188 @ SY189 ) )
| ( member @ SV12 @ SY188 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[203]) ).
thf(302,plain,
! [SV13: $i] :
( ( ! [SY190: $i,SY191: $i,SY192: $i] :
( ~ ( member @ ( ordered_pair @ SV13 @ SY190 ) @ ( cross_product @ SY191 @ SY192 ) )
| ( member @ SY190 @ SY192 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[204]) ).
thf(303,plain,
! [SV14: $i] :
( ( ! [SY193: $i,SY194: $i,SY195: $i] :
( ~ ( member @ SV14 @ SY193 )
| ~ ( member @ SY194 @ SY195 )
| ( member @ ( ordered_pair @ SV14 @ SY194 ) @ ( cross_product @ SY193 @ SY195 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[205]) ).
thf(304,plain,
! [SV15: $i] :
( ( ! [SY196: $i,SY197: $i] :
( ~ ( member @ SV15 @ ( cross_product @ SY196 @ SY197 ) )
| ( ( ordered_pair @ ( first @ SV15 ) @ ( second @ SV15 ) )
= SV15 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[206]) ).
thf(305,plain,
! [SV16: $i] :
( ( ! [SY198: $i] :
( ~ ( member @ ( ordered_pair @ SV16 @ SY198 ) @ element_relation )
| ( member @ SV16 @ SY198 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[208]) ).
thf(306,plain,
! [SV17: $i] :
( ( ! [SY199: $i] :
( ~ ( member @ ( ordered_pair @ SV17 @ SY199 ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ SV17 @ SY199 )
| ( member @ ( ordered_pair @ SV17 @ SY199 ) @ element_relation ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[209]) ).
thf(307,plain,
! [SV18: $i] :
( ( ! [SY200: $i,SY201: $i] :
( ~ ( member @ SV18 @ ( intersection @ SY200 @ SY201 ) )
| ( member @ SV18 @ SY200 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[210]) ).
thf(308,plain,
! [SV19: $i] :
( ( ! [SY202: $i,SY203: $i] :
( ~ ( member @ SV19 @ ( intersection @ SY202 @ SY203 ) )
| ( member @ SV19 @ SY203 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[211]) ).
thf(309,plain,
! [SV20: $i] :
( ( ! [SY204: $i,SY205: $i] :
( ~ ( member @ SV20 @ SY204 )
| ~ ( member @ SV20 @ SY205 )
| ( member @ SV20 @ ( intersection @ SY204 @ SY205 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[212]) ).
thf(310,plain,
! [SV21: $i] :
( ( ! [SY206: $i] :
( ~ ( member @ SV21 @ ( complement @ SY206 ) )
| ~ ( member @ SV21 @ SY206 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[213]) ).
thf(311,plain,
! [SV22: $i] :
( ( ! [SY207: $i] :
( ~ ( member @ SV22 @ universal_class )
| ( member @ SV22 @ ( complement @ SY207 ) )
| ( member @ SV22 @ SY207 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[214]) ).
thf(312,plain,
! [SV23: $i] :
( ( ! [SY208: $i] :
( ( complement @ ( intersection @ ( complement @ SV23 ) @ ( complement @ SY208 ) ) )
= ( union @ SV23 @ SY208 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[215]) ).
thf(313,plain,
! [SV24: $i] :
( ( ! [SY209: $i] :
( ( intersection @ ( complement @ ( intersection @ SV24 @ SY209 ) ) @ ( complement @ ( intersection @ ( complement @ SV24 ) @ ( complement @ SY209 ) ) ) )
= ( symmetric_difference @ SV24 @ SY209 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[216]) ).
thf(314,plain,
! [SV25: $i] :
( ( ! [SY210: $i,SY211: $i] :
( ( intersection @ SV25 @ ( cross_product @ SY210 @ SY211 ) )
= ( restrict @ SV25 @ SY210 @ SY211 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[217]) ).
thf(315,plain,
! [SV26: $i] :
( ( ! [SY212: $i,SY213: $i] :
( ( intersection @ ( cross_product @ SV26 @ SY212 ) @ SY213 )
= ( restrict @ SY213 @ SV26 @ SY212 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[218]) ).
thf(316,plain,
! [SV27: $i] :
( ( ! [SY214: $i] :
( ~ ( member @ SV27 @ universal_class )
| ( ( restrict @ SY214 @ ( singleton @ SV27 ) @ universal_class )
= null_class )
| ( member @ SV27 @ ( domain_of @ SY214 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[220]) ).
thf(317,plain,
! [SV28: $i] :
( ( subclass @ ( rotate @ SV28 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[221]) ).
thf(318,plain,
! [SV29: $i] :
( ( ! [SY215: $i,SY216: $i,SY217: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV29 @ SY215 ) @ SY216 ) @ ( rotate @ SY217 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY215 @ SY216 ) @ SV29 ) @ SY217 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[222]) ).
thf(319,plain,
! [SV30: $i] :
( ( ! [SY218: $i,SY219: $i,SY220: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV30 @ SY218 ) @ SY219 ) @ SY220 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SY219 @ SV30 ) @ SY218 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY219 @ SV30 ) @ SY218 ) @ ( rotate @ SY220 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[223]) ).
thf(320,plain,
! [SV31: $i] :
( ( subclass @ ( flip @ SV31 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[224]) ).
thf(321,plain,
! [SV32: $i] :
( ( ! [SY221: $i,SY222: $i,SY223: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SY221 ) @ SY222 ) @ ( flip @ SY223 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY221 @ SV32 ) @ SY222 ) @ SY223 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[225]) ).
thf(322,plain,
! [SV33: $i] :
( ( ! [SY224: $i,SY225: $i,SY226: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SY224 ) @ SY225 ) @ SY226 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SY224 @ SV33 ) @ SY225 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY224 @ SV33 ) @ SY225 ) @ ( flip @ SY226 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[226]) ).
thf(323,plain,
! [SV34: $i] :
( ( ( domain_of @ ( flip @ ( cross_product @ SV34 @ universal_class ) ) )
= ( inverse @ SV34 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[227]) ).
thf(324,plain,
! [SV35: $i] :
( ( ( domain_of @ ( inverse @ SV35 ) )
= ( range_of @ SV35 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[228]) ).
thf(325,plain,
! [SV36: $i] :
( ( ! [SY227: $i,SY228: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ SV36 @ SY227 @ ( singleton @ SY228 ) ) @ null_class ) )
= ( domain @ SV36 @ SY227 @ SY228 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[229]) ).
thf(326,plain,
! [SV37: $i] :
( ( ! [SY229: $i,SY230: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ SV37 @ ( singleton @ SY229 ) @ SY230 ) @ null_class ) )
= ( range @ SV37 @ SY229 @ SY230 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[230]) ).
thf(327,plain,
! [SV38: $i] :
( ( ! [SY231: $i] :
( ( range_of @ ( restrict @ SV38 @ SY231 @ universal_class ) )
= ( image @ SV38 @ SY231 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[231]) ).
thf(328,plain,
! [SV39: $i] :
( ( ( union @ SV39 @ ( singleton @ SV39 ) )
= ( successor @ SV39 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[232]) ).
thf(329,plain,
! [SV40: $i] :
( ( ! [SY232: $i] :
( ~ ( member @ ( ordered_pair @ SV40 @ SY232 ) @ successor_relation )
| ( ( successor @ SV40 )
= SY232 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[234]) ).
thf(330,plain,
! [SV41: $i] :
( ( ~ ( inductive @ SV41 )
| ( member @ null_class @ SV41 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[236]) ).
thf(331,plain,
! [SV42: $i] :
( ( ~ ( inductive @ SV42 )
| ( subclass @ ( image @ successor_relation @ SV42 ) @ SV42 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[237]) ).
thf(332,plain,
! [SV43: $i] :
( ( ~ ( member @ null_class @ SV43 )
| ~ ( subclass @ ( image @ successor_relation @ SV43 ) @ SV43 )
| ( inductive @ SV43 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[238]) ).
thf(333,plain,
! [SV44: $i] :
( ( ~ ( inductive @ SV44 )
| ( subclass @ omega @ SV44 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[240]) ).
thf(334,plain,
! [SV45: $i] :
( ( ( domain_of @ ( restrict @ element_relation @ universal_class @ SV45 ) )
= ( sum_class @ SV45 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[242]) ).
thf(335,plain,
! [SV46: $i] :
( ( ~ ( member @ SV46 @ universal_class )
| ( member @ ( sum_class @ SV46 ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[243]) ).
thf(336,plain,
! [SV47: $i] :
( ( ( complement @ ( image @ element_relation @ ( complement @ SV47 ) ) )
= ( power_class @ SV47 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[244]) ).
thf(337,plain,
! [SV48: $i] :
( ( ~ ( member @ SV48 @ universal_class )
| ( member @ ( power_class @ SV48 ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[245]) ).
thf(338,plain,
! [SV49: $i] :
( ( ! [SY233: $i] : ( subclass @ ( compose @ SV49 @ SY233 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[246]) ).
thf(339,plain,
! [SV50: $i] :
( ( ! [SY234: $i,SY235: $i,SY236: $i] :
( ~ ( member @ ( ordered_pair @ SV50 @ SY234 ) @ ( compose @ SY235 @ SY236 ) )
| ( member @ SY234 @ ( image @ SY235 @ ( image @ SY236 @ ( singleton @ SV50 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[247]) ).
thf(340,plain,
! [SV51: $i] :
( ( ! [SY237: $i,SY238: $i,SY239: $i] :
( ~ ( member @ SV51 @ ( image @ SY237 @ ( image @ SY238 @ ( singleton @ SY239 ) ) ) )
| ~ ( member @ ( ordered_pair @ SY239 @ SV51 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SY239 @ SV51 ) @ ( compose @ SY237 @ SY238 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[248]) ).
thf(341,plain,
! [SV52: $i] :
( ( ~ ( single_valued_class @ SV52 )
| ( subclass @ ( compose @ SV52 @ ( inverse @ SV52 ) ) @ identity_relation ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[249]) ).
thf(342,plain,
! [SV53: $i] :
( ( ~ ( subclass @ ( compose @ SV53 @ ( inverse @ SV53 ) ) @ identity_relation )
| ( single_valued_class @ SV53 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[250]) ).
thf(343,plain,
! [SV54: $i] :
( ( ~ ( function @ SV54 )
| ( subclass @ SV54 @ ( cross_product @ universal_class @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[251]) ).
thf(344,plain,
! [SV55: $i] :
( ( ~ ( function @ SV55 )
| ( subclass @ ( compose @ SV55 @ ( inverse @ SV55 ) ) @ identity_relation ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[252]) ).
thf(345,plain,
! [SV56: $i] :
( ( ~ ( subclass @ SV56 @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( subclass @ ( compose @ SV56 @ ( inverse @ SV56 ) ) @ identity_relation )
| ( function @ SV56 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[253]) ).
thf(346,plain,
! [SV57: $i] :
( ( ! [SY240: $i] :
( ~ ( function @ SV57 )
| ~ ( member @ SY240 @ universal_class )
| ( member @ ( image @ SV57 @ SY240 ) @ universal_class ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[254]) ).
thf(347,plain,
! [SV58: $i] :
( ( ( SV58 = null_class )
| ( member @ ( regular @ SV58 ) @ SV58 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[255]) ).
thf(348,plain,
! [SV59: $i] :
( ( ( SV59 = null_class )
| ( ( intersection @ SV59 @ ( regular @ SV59 ) )
= null_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[256]) ).
thf(349,plain,
! [SV60: $i] :
( ( ! [SY241: $i] :
( ( sum_class @ ( image @ SV60 @ ( singleton @ SY241 ) ) )
= ( apply @ SV60 @ SY241 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[257]) ).
thf(350,plain,
! [SV61: $i] :
( ( ~ ( member @ SV61 @ universal_class )
| ( SV61 = null_class )
| ( member @ ( apply @ choice @ SV61 ) @ SV61 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[259]) ).
thf(351,plain,
! [SV62: $i] :
( ( ~ ( one_to_one @ SV62 )
| ( function @ SV62 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[260]) ).
thf(352,plain,
! [SV63: $i] :
( ( ~ ( one_to_one @ SV63 )
| ( function @ ( inverse @ SV63 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[261]) ).
thf(353,plain,
! [SV64: $i] :
( ( ~ ( function @ ( inverse @ SV64 ) )
| ~ ( function @ SV64 )
| ( one_to_one @ SV64 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[262]) ).
thf(354,plain,
! [SV65: $i] :
( ( ( complement @ ( domain_of @ ( intersection @ SV65 @ identity_relation ) ) )
= ( diagonalise @ SV65 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[265]) ).
thf(355,plain,
! [SV66: $i] :
( ( ( intersection @ ( domain_of @ SV66 ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ SV66 ) ) )
= ( cantor @ SV66 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[266]) ).
thf(356,plain,
! [SV67: $i] :
( ( ~ ( operation @ SV67 )
| ( function @ SV67 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[267]) ).
thf(357,plain,
! [SV68: $i] :
( ( ~ ( operation @ SV68 )
| ( ( cross_product @ ( domain_of @ ( domain_of @ SV68 ) ) @ ( domain_of @ ( domain_of @ SV68 ) ) )
= ( domain_of @ SV68 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[268]) ).
thf(358,plain,
! [SV69: $i] :
( ( ~ ( operation @ SV69 )
| ( subclass @ ( range_of @ SV69 ) @ ( domain_of @ ( domain_of @ SV69 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[269]) ).
thf(359,plain,
! [SV70: $i] :
( ( ! [SY242: $i,SY243: $i] :
( ~ ( compatible @ SV70 @ SY242 @ SY243 )
| ( function @ SV70 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[271]) ).
thf(360,plain,
! [SV71: $i] :
( ( ! [SY244: $i,SY245: $i] :
( ~ ( compatible @ SV71 @ SY244 @ SY245 )
| ( ( domain_of @ ( domain_of @ SY244 ) )
= ( domain_of @ SV71 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[272]) ).
thf(361,plain,
! [SV72: $i] :
( ( ! [SY246: $i,SY247: $i] :
( ~ ( compatible @ SV72 @ SY246 @ SY247 )
| ( subclass @ ( range_of @ SV72 ) @ ( domain_of @ ( domain_of @ SY247 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[273]) ).
thf(362,plain,
! [SV73: $i] :
( ( ! [SY248: $i,SY249: $i] :
( ~ ( homomorphism @ SV73 @ SY248 @ SY249 )
| ( operation @ SY248 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[275]) ).
thf(363,plain,
! [SV74: $i] :
( ( ! [SY250: $i,SY251: $i] :
( ~ ( homomorphism @ SV74 @ SY250 @ SY251 )
| ( operation @ SY251 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[276]) ).
thf(364,plain,
! [SV75: $i] :
( ( ! [SY252: $i,SY253: $i] :
( ~ ( homomorphism @ SV75 @ SY252 @ SY253 )
| ( compatible @ SV75 @ SY252 @ SY253 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[277]) ).
thf(365,plain,
! [SV76: $i] :
( ( ! [SY254: $i,SY255: $i,SY256: $i,SY257: $i] :
( ~ ( homomorphism @ SV76 @ SY254 @ SY255 )
| ~ ( member @ ( ordered_pair @ SY256 @ SY257 ) @ ( domain_of @ SY254 ) )
| ( ( apply @ SY255 @ ( ordered_pair @ ( apply @ SV76 @ SY256 ) @ ( apply @ SV76 @ SY257 ) ) )
= ( apply @ SV76 @ ( apply @ SY254 @ ( ordered_pair @ SY256 @ SY257 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[278]) ).
thf(366,plain,
! [SV77: $i] :
( ( ! [SY258: $i,SY259: $i] :
( ~ ( operation @ SV77 )
| ~ ( operation @ SY258 )
| ~ ( compatible @ SY259 @ SV77 @ SY258 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SY259 @ SV77 @ SY258 ) @ ( not_homomorphism2 @ SY259 @ SV77 @ SY258 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SY259 @ SV77 @ SY258 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[279]) ).
thf(367,plain,
$false = $false,
inference(extcnf_not_pos,[status(thm)],[281]) ).
thf(368,plain,
! [SV78: $i] :
( ( ! [SY260: $i,SY261: $i] :
( ~ ( function @ SV78 )
| ( ( domain_of @ ( domain_of @ SY260 ) )
!= ( domain_of @ SV78 ) )
| ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SY261 ) ) )
| ( compatible @ SV78 @ SY260 @ SY261 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[282]) ).
thf(369,plain,
! [SV79: $i] :
( ( ! [SY262: $i] :
( ( SV79 != SY262 )
| ( subclass @ SV79 @ SY262 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[283]) ).
thf(370,plain,
! [SV80: $i] :
( ( ! [SY263: $i] :
( ( SV80 != SY263 )
| ( subclass @ SY263 @ SV80 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[284]) ).
thf(371,plain,
! [SV81: $i] :
( ( ~ ( function @ SV81 )
| ( ( cross_product @ ( domain_of @ ( domain_of @ SV81 ) ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
!= ( domain_of @ SV81 ) )
| ~ ( subclass @ ( range_of @ SV81 ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
| ( operation @ SV81 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[285]) ).
thf(372,plain,
! [SV82: $i] :
( ( ! [SY264: $i,SY265: $i] :
( ~ ( operation @ SV82 )
| ~ ( operation @ SY264 )
| ~ ( compatible @ SY265 @ SV82 @ SY264 )
| ( ( apply @ SY264 @ ( ordered_pair @ ( apply @ SY265 @ ( not_homomorphism1 @ SY265 @ SV82 @ SY264 ) ) @ ( apply @ SY265 @ ( not_homomorphism2 @ SY265 @ SV82 @ SY264 ) ) ) )
!= ( apply @ SY265 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SY265 @ SV82 @ SY264 ) @ ( not_homomorphism2 @ SY265 @ SV82 @ SY264 ) ) ) ) )
| ( homomorphism @ SY265 @ SV82 @ SY264 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[286]) ).
thf(373,plain,
! [SV83: $i] :
( ( ! [SY266: $i] :
( ( ( successor @ SV83 )
!= SY266 )
| ~ ( member @ ( ordered_pair @ SV83 @ SY266 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV83 @ SY266 ) @ successor_relation ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[287]) ).
thf(374,plain,
! [SV84: $i] :
( ( ! [SY267: $i] :
( ( ( restrict @ SV84 @ ( singleton @ SY267 ) @ universal_class )
!= null_class )
| ~ ( member @ SY267 @ ( domain_of @ SV84 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[288]) ).
thf(375,plain,
! [SV85: $i,SV1: $i] :
( ( ! [SY268: $i] :
( ~ ( subclass @ SV1 @ SV85 )
| ~ ( member @ SY268 @ SV1 )
| ( member @ SY268 @ SV85 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[290]) ).
thf(376,plain,
! [SV86: $i,SV2: $i] :
( ( ( member @ ( not_subclass_element @ SV2 @ SV86 ) @ SV2 )
| ( subclass @ SV2 @ SV86 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[291]) ).
thf(377,plain,
! [SV87: $i,SV3: $i] :
( ( ~ ( member @ ( not_subclass_element @ SV3 @ SV87 ) @ SV87 )
| ( subclass @ SV3 @ SV87 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[292]) ).
thf(378,plain,
! [SV88: $i,SV5: $i] :
( ( ~ ( subclass @ SV5 @ SV88 )
| ~ ( subclass @ SV88 @ SV5 )
| ( SV5 = SV88 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[294]) ).
thf(379,plain,
! [SV89: $i,SV6: $i] :
( ( ! [SY269: $i] :
( ~ ( member @ SV6 @ ( unordered_pair @ SV89 @ SY269 ) )
| ( SV6 = SV89 )
| ( SV6 = SY269 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[295]) ).
thf(380,plain,
! [SV90: $i,SV7: $i] :
( ( ~ ( member @ SV7 @ universal_class )
| ( member @ SV7 @ ( unordered_pair @ SV7 @ SV90 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[296]) ).
thf(381,plain,
! [SV91: $i,SV8: $i] :
( ( ~ ( member @ SV8 @ universal_class )
| ( member @ SV8 @ ( unordered_pair @ SV91 @ SV8 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[297]) ).
thf(382,plain,
! [SV92: $i,SV9: $i] :
( ( member @ ( unordered_pair @ SV9 @ SV92 ) @ universal_class )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[298]) ).
thf(383,plain,
! [SV93: $i,SV11: $i] :
( ( ( unordered_pair @ ( singleton @ SV11 ) @ ( unordered_pair @ SV11 @ ( singleton @ SV93 ) ) )
= ( ordered_pair @ SV11 @ SV93 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[300]) ).
thf(384,plain,
! [SV94: $i,SV12: $i] :
( ( ! [SY270: $i,SY271: $i] :
( ~ ( member @ ( ordered_pair @ SV12 @ SV94 ) @ ( cross_product @ SY270 @ SY271 ) )
| ( member @ SV12 @ SY270 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[301]) ).
thf(385,plain,
! [SV95: $i,SV13: $i] :
( ( ! [SY272: $i,SY273: $i] :
( ~ ( member @ ( ordered_pair @ SV13 @ SV95 ) @ ( cross_product @ SY272 @ SY273 ) )
| ( member @ SV95 @ SY273 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[302]) ).
thf(386,plain,
! [SV96: $i,SV14: $i] :
( ( ! [SY274: $i,SY275: $i] :
( ~ ( member @ SV14 @ SV96 )
| ~ ( member @ SY274 @ SY275 )
| ( member @ ( ordered_pair @ SV14 @ SY274 ) @ ( cross_product @ SV96 @ SY275 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[303]) ).
thf(387,plain,
! [SV97: $i,SV15: $i] :
( ( ! [SY276: $i] :
( ~ ( member @ SV15 @ ( cross_product @ SV97 @ SY276 ) )
| ( ( ordered_pair @ ( first @ SV15 ) @ ( second @ SV15 ) )
= SV15 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[304]) ).
thf(388,plain,
! [SV98: $i,SV16: $i] :
( ( ~ ( member @ ( ordered_pair @ SV16 @ SV98 ) @ element_relation )
| ( member @ SV16 @ SV98 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[305]) ).
thf(389,plain,
! [SV99: $i,SV17: $i] :
( ( ~ ( member @ ( ordered_pair @ SV17 @ SV99 ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ SV17 @ SV99 )
| ( member @ ( ordered_pair @ SV17 @ SV99 ) @ element_relation ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[306]) ).
thf(390,plain,
! [SV100: $i,SV18: $i] :
( ( ! [SY277: $i] :
( ~ ( member @ SV18 @ ( intersection @ SV100 @ SY277 ) )
| ( member @ SV18 @ SV100 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[307]) ).
thf(391,plain,
! [SV101: $i,SV19: $i] :
( ( ! [SY278: $i] :
( ~ ( member @ SV19 @ ( intersection @ SV101 @ SY278 ) )
| ( member @ SV19 @ SY278 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[308]) ).
thf(392,plain,
! [SV102: $i,SV20: $i] :
( ( ! [SY279: $i] :
( ~ ( member @ SV20 @ SV102 )
| ~ ( member @ SV20 @ SY279 )
| ( member @ SV20 @ ( intersection @ SV102 @ SY279 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[309]) ).
thf(393,plain,
! [SV103: $i,SV21: $i] :
( ( ~ ( member @ SV21 @ ( complement @ SV103 ) )
| ~ ( member @ SV21 @ SV103 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[310]) ).
thf(394,plain,
! [SV104: $i,SV22: $i] :
( ( ~ ( member @ SV22 @ universal_class )
| ( member @ SV22 @ ( complement @ SV104 ) )
| ( member @ SV22 @ SV104 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[311]) ).
thf(395,plain,
! [SV105: $i,SV23: $i] :
( ( ( complement @ ( intersection @ ( complement @ SV23 ) @ ( complement @ SV105 ) ) )
= ( union @ SV23 @ SV105 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[312]) ).
thf(396,plain,
! [SV106: $i,SV24: $i] :
( ( ( intersection @ ( complement @ ( intersection @ SV24 @ SV106 ) ) @ ( complement @ ( intersection @ ( complement @ SV24 ) @ ( complement @ SV106 ) ) ) )
= ( symmetric_difference @ SV24 @ SV106 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[313]) ).
thf(397,plain,
! [SV107: $i,SV25: $i] :
( ( ! [SY280: $i] :
( ( intersection @ SV25 @ ( cross_product @ SV107 @ SY280 ) )
= ( restrict @ SV25 @ SV107 @ SY280 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[314]) ).
thf(398,plain,
! [SV108: $i,SV26: $i] :
( ( ! [SY281: $i] :
( ( intersection @ ( cross_product @ SV26 @ SV108 ) @ SY281 )
= ( restrict @ SY281 @ SV26 @ SV108 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[315]) ).
thf(399,plain,
! [SV109: $i,SV27: $i] :
( ( ~ ( member @ SV27 @ universal_class )
| ( ( restrict @ SV109 @ ( singleton @ SV27 ) @ universal_class )
= null_class )
| ( member @ SV27 @ ( domain_of @ SV109 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[316]) ).
thf(400,plain,
! [SV110: $i,SV29: $i] :
( ( ! [SY282: $i,SY283: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV29 @ SV110 ) @ SY282 ) @ ( rotate @ SY283 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV110 @ SY282 ) @ SV29 ) @ SY283 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[318]) ).
thf(401,plain,
! [SV111: $i,SV30: $i] :
( ( ! [SY284: $i,SY285: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV30 @ SV111 ) @ SY284 ) @ SY285 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SY284 @ SV30 ) @ SV111 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SY284 @ SV30 ) @ SV111 ) @ ( rotate @ SY285 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[319]) ).
thf(402,plain,
! [SV112: $i,SV32: $i] :
( ( ! [SY286: $i,SY287: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV112 ) @ SY286 ) @ ( flip @ SY287 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV112 @ SV32 ) @ SY286 ) @ SY287 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[321]) ).
thf(403,plain,
! [SV113: $i,SV33: $i] :
( ( ! [SY288: $i,SY289: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV113 ) @ SY288 ) @ SY289 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SY288 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SY288 ) @ ( flip @ SY289 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[322]) ).
thf(404,plain,
! [SV114: $i,SV36: $i] :
( ( ! [SY290: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ SV36 @ SV114 @ ( singleton @ SY290 ) ) @ null_class ) )
= ( domain @ SV36 @ SV114 @ SY290 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[325]) ).
thf(405,plain,
! [SV115: $i,SV37: $i] :
( ( ! [SY291: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ SV37 @ ( singleton @ SV115 ) @ SY291 ) @ null_class ) )
= ( range @ SV37 @ SV115 @ SY291 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[326]) ).
thf(406,plain,
! [SV116: $i,SV38: $i] :
( ( ( range_of @ ( restrict @ SV38 @ SV116 @ universal_class ) )
= ( image @ SV38 @ SV116 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[327]) ).
thf(407,plain,
! [SV117: $i,SV40: $i] :
( ( ~ ( member @ ( ordered_pair @ SV40 @ SV117 ) @ successor_relation )
| ( ( successor @ SV40 )
= SV117 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[329]) ).
thf(408,plain,
! [SV41: $i] :
( ( ( ~ ( inductive @ SV41 ) )
= $true )
| ( ( member @ null_class @ SV41 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[330]) ).
thf(409,plain,
! [SV42: $i] :
( ( ( ~ ( inductive @ SV42 ) )
= $true )
| ( ( subclass @ ( image @ successor_relation @ SV42 ) @ SV42 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[331]) ).
thf(410,plain,
! [SV43: $i] :
( ( ( ~ ( member @ null_class @ SV43 ) )
= $true )
| ( ( ~ ( subclass @ ( image @ successor_relation @ SV43 ) @ SV43 )
| ( inductive @ SV43 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[332]) ).
thf(411,plain,
! [SV44: $i] :
( ( ( ~ ( inductive @ SV44 ) )
= $true )
| ( ( subclass @ omega @ SV44 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[333]) ).
thf(412,plain,
! [SV46: $i] :
( ( ( ~ ( member @ SV46 @ universal_class ) )
= $true )
| ( ( member @ ( sum_class @ SV46 ) @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[335]) ).
thf(413,plain,
! [SV48: $i] :
( ( ( ~ ( member @ SV48 @ universal_class ) )
= $true )
| ( ( member @ ( power_class @ SV48 ) @ universal_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[337]) ).
thf(414,plain,
! [SV118: $i,SV49: $i] :
( ( subclass @ ( compose @ SV49 @ SV118 ) @ ( cross_product @ universal_class @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[338]) ).
thf(415,plain,
! [SV119: $i,SV50: $i] :
( ( ! [SY292: $i,SY293: $i] :
( ~ ( member @ ( ordered_pair @ SV50 @ SV119 ) @ ( compose @ SY292 @ SY293 ) )
| ( member @ SV119 @ ( image @ SY292 @ ( image @ SY293 @ ( singleton @ SV50 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[339]) ).
thf(416,plain,
! [SV120: $i,SV51: $i] :
( ( ! [SY294: $i,SY295: $i] :
( ~ ( member @ SV51 @ ( image @ SV120 @ ( image @ SY294 @ ( singleton @ SY295 ) ) ) )
| ~ ( member @ ( ordered_pair @ SY295 @ SV51 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SY295 @ SV51 ) @ ( compose @ SV120 @ SY294 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[340]) ).
thf(417,plain,
! [SV52: $i] :
( ( ( ~ ( single_valued_class @ SV52 ) )
= $true )
| ( ( subclass @ ( compose @ SV52 @ ( inverse @ SV52 ) ) @ identity_relation )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[341]) ).
thf(418,plain,
! [SV53: $i] :
( ( ( ~ ( subclass @ ( compose @ SV53 @ ( inverse @ SV53 ) ) @ identity_relation ) )
= $true )
| ( ( single_valued_class @ SV53 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[342]) ).
thf(419,plain,
! [SV54: $i] :
( ( ( ~ ( function @ SV54 ) )
= $true )
| ( ( subclass @ SV54 @ ( cross_product @ universal_class @ universal_class ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[343]) ).
thf(420,plain,
! [SV55: $i] :
( ( ( ~ ( function @ SV55 ) )
= $true )
| ( ( subclass @ ( compose @ SV55 @ ( inverse @ SV55 ) ) @ identity_relation )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[344]) ).
thf(421,plain,
! [SV56: $i] :
( ( ( ~ ( subclass @ SV56 @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( ~ ( subclass @ ( compose @ SV56 @ ( inverse @ SV56 ) ) @ identity_relation )
| ( function @ SV56 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[345]) ).
thf(422,plain,
! [SV121: $i,SV57: $i] :
( ( ~ ( function @ SV57 )
| ~ ( member @ SV121 @ universal_class )
| ( member @ ( image @ SV57 @ SV121 ) @ universal_class ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[346]) ).
thf(423,plain,
! [SV58: $i] :
( ( ( SV58 = null_class )
= $true )
| ( ( member @ ( regular @ SV58 ) @ SV58 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[347]) ).
thf(424,plain,
! [SV59: $i] :
( ( ( SV59 = null_class )
= $true )
| ( ( ( intersection @ SV59 @ ( regular @ SV59 ) )
= null_class )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[348]) ).
thf(425,plain,
! [SV122: $i,SV60: $i] :
( ( ( sum_class @ ( image @ SV60 @ ( singleton @ SV122 ) ) )
= ( apply @ SV60 @ SV122 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[349]) ).
thf(426,plain,
! [SV61: $i] :
( ( ( ~ ( member @ SV61 @ universal_class ) )
= $true )
| ( ( ( SV61 = null_class )
| ( member @ ( apply @ choice @ SV61 ) @ SV61 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[350]) ).
thf(427,plain,
! [SV62: $i] :
( ( ( ~ ( one_to_one @ SV62 ) )
= $true )
| ( ( function @ SV62 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[351]) ).
thf(428,plain,
! [SV63: $i] :
( ( ( ~ ( one_to_one @ SV63 ) )
= $true )
| ( ( function @ ( inverse @ SV63 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[352]) ).
thf(429,plain,
! [SV64: $i] :
( ( ( ~ ( function @ ( inverse @ SV64 ) ) )
= $true )
| ( ( ~ ( function @ SV64 )
| ( one_to_one @ SV64 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[353]) ).
thf(430,plain,
! [SV67: $i] :
( ( ( ~ ( operation @ SV67 ) )
= $true )
| ( ( function @ SV67 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[356]) ).
thf(431,plain,
! [SV68: $i] :
( ( ( ~ ( operation @ SV68 ) )
= $true )
| ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV68 ) ) @ ( domain_of @ ( domain_of @ SV68 ) ) )
= ( domain_of @ SV68 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[357]) ).
thf(432,plain,
! [SV69: $i] :
( ( ( ~ ( operation @ SV69 ) )
= $true )
| ( ( subclass @ ( range_of @ SV69 ) @ ( domain_of @ ( domain_of @ SV69 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[358]) ).
thf(433,plain,
! [SV123: $i,SV70: $i] :
( ( ! [SY296: $i] :
( ~ ( compatible @ SV70 @ SV123 @ SY296 )
| ( function @ SV70 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[359]) ).
thf(434,plain,
! [SV124: $i,SV71: $i] :
( ( ! [SY297: $i] :
( ~ ( compatible @ SV71 @ SV124 @ SY297 )
| ( ( domain_of @ ( domain_of @ SV124 ) )
= ( domain_of @ SV71 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[360]) ).
thf(435,plain,
! [SV125: $i,SV72: $i] :
( ( ! [SY298: $i] :
( ~ ( compatible @ SV72 @ SV125 @ SY298 )
| ( subclass @ ( range_of @ SV72 ) @ ( domain_of @ ( domain_of @ SY298 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[361]) ).
thf(436,plain,
! [SV126: $i,SV73: $i] :
( ( ! [SY299: $i] :
( ~ ( homomorphism @ SV73 @ SV126 @ SY299 )
| ( operation @ SV126 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[362]) ).
thf(437,plain,
! [SV127: $i,SV74: $i] :
( ( ! [SY300: $i] :
( ~ ( homomorphism @ SV74 @ SV127 @ SY300 )
| ( operation @ SY300 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[363]) ).
thf(438,plain,
! [SV128: $i,SV75: $i] :
( ( ! [SY301: $i] :
( ~ ( homomorphism @ SV75 @ SV128 @ SY301 )
| ( compatible @ SV75 @ SV128 @ SY301 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[364]) ).
thf(439,plain,
! [SV129: $i,SV76: $i] :
( ( ! [SY302: $i,SY303: $i,SY304: $i] :
( ~ ( homomorphism @ SV76 @ SV129 @ SY302 )
| ~ ( member @ ( ordered_pair @ SY303 @ SY304 ) @ ( domain_of @ SV129 ) )
| ( ( apply @ SY302 @ ( ordered_pair @ ( apply @ SV76 @ SY303 ) @ ( apply @ SV76 @ SY304 ) ) )
= ( apply @ SV76 @ ( apply @ SV129 @ ( ordered_pair @ SY303 @ SY304 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[365]) ).
thf(440,plain,
! [SV130: $i,SV77: $i] :
( ( ! [SY305: $i] :
( ~ ( operation @ SV77 )
| ~ ( operation @ SV130 )
| ~ ( compatible @ SY305 @ SV77 @ SV130 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SY305 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SY305 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SY305 @ SV77 @ SV130 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[366]) ).
thf(441,plain,
! [SV131: $i,SV78: $i] :
( ( ! [SY306: $i] :
( ~ ( function @ SV78 )
| ( ( domain_of @ ( domain_of @ SV131 ) )
!= ( domain_of @ SV78 ) )
| ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SY306 ) ) )
| ( compatible @ SV78 @ SV131 @ SY306 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[368]) ).
thf(442,plain,
! [SV132: $i,SV79: $i] :
( ( ( SV79 != SV132 )
| ( subclass @ SV79 @ SV132 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[369]) ).
thf(443,plain,
! [SV133: $i,SV80: $i] :
( ( ( SV80 != SV133 )
| ( subclass @ SV133 @ SV80 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[370]) ).
thf(444,plain,
! [SV81: $i] :
( ( ( ~ ( function @ SV81 ) )
= $true )
| ( ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV81 ) ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
!= ( domain_of @ SV81 ) )
| ~ ( subclass @ ( range_of @ SV81 ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
| ( operation @ SV81 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[371]) ).
thf(445,plain,
! [SV134: $i,SV82: $i] :
( ( ! [SY307: $i] :
( ~ ( operation @ SV82 )
| ~ ( operation @ SV134 )
| ~ ( compatible @ SY307 @ SV82 @ SV134 )
| ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SY307 @ ( not_homomorphism1 @ SY307 @ SV82 @ SV134 ) ) @ ( apply @ SY307 @ ( not_homomorphism2 @ SY307 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SY307 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SY307 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SY307 @ SV82 @ SV134 ) ) ) ) )
| ( homomorphism @ SY307 @ SV82 @ SV134 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[372]) ).
thf(446,plain,
! [SV135: $i,SV83: $i] :
( ( ( ( successor @ SV83 )
!= SV135 )
| ~ ( member @ ( ordered_pair @ SV83 @ SV135 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV83 @ SV135 ) @ successor_relation ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[373]) ).
thf(447,plain,
! [SV136: $i,SV84: $i] :
( ( ( ( restrict @ SV84 @ ( singleton @ SV136 ) @ universal_class )
!= null_class )
| ~ ( member @ SV136 @ ( domain_of @ SV84 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[374]) ).
thf(448,plain,
! [SV137: $i,SV85: $i,SV1: $i] :
( ( ~ ( subclass @ SV1 @ SV85 )
| ~ ( member @ SV137 @ SV1 )
| ( member @ SV137 @ SV85 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[375]) ).
thf(449,plain,
! [SV86: $i,SV2: $i] :
( ( ( member @ ( not_subclass_element @ SV2 @ SV86 ) @ SV2 )
= $true )
| ( ( subclass @ SV2 @ SV86 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[376]) ).
thf(450,plain,
! [SV87: $i,SV3: $i] :
( ( ( ~ ( member @ ( not_subclass_element @ SV3 @ SV87 ) @ SV87 ) )
= $true )
| ( ( subclass @ SV3 @ SV87 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[377]) ).
thf(451,plain,
! [SV88: $i,SV5: $i] :
( ( ( ~ ( subclass @ SV5 @ SV88 ) )
= $true )
| ( ( ~ ( subclass @ SV88 @ SV5 )
| ( SV5 = SV88 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[378]) ).
thf(452,plain,
! [SV138: $i,SV89: $i,SV6: $i] :
( ( ~ ( member @ SV6 @ ( unordered_pair @ SV89 @ SV138 ) )
| ( SV6 = SV89 )
| ( SV6 = SV138 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[379]) ).
thf(453,plain,
! [SV90: $i,SV7: $i] :
( ( ( ~ ( member @ SV7 @ universal_class ) )
= $true )
| ( ( member @ SV7 @ ( unordered_pair @ SV7 @ SV90 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[380]) ).
thf(454,plain,
! [SV91: $i,SV8: $i] :
( ( ( ~ ( member @ SV8 @ universal_class ) )
= $true )
| ( ( member @ SV8 @ ( unordered_pair @ SV91 @ SV8 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[381]) ).
thf(455,plain,
! [SV139: $i,SV94: $i,SV12: $i] :
( ( ! [SY308: $i] :
( ~ ( member @ ( ordered_pair @ SV12 @ SV94 ) @ ( cross_product @ SV139 @ SY308 ) )
| ( member @ SV12 @ SV139 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[384]) ).
thf(456,plain,
! [SV140: $i,SV95: $i,SV13: $i] :
( ( ! [SY309: $i] :
( ~ ( member @ ( ordered_pair @ SV13 @ SV95 ) @ ( cross_product @ SV140 @ SY309 ) )
| ( member @ SV95 @ SY309 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[385]) ).
thf(457,plain,
! [SV141: $i,SV96: $i,SV14: $i] :
( ( ! [SY310: $i] :
( ~ ( member @ SV14 @ SV96 )
| ~ ( member @ SV141 @ SY310 )
| ( member @ ( ordered_pair @ SV14 @ SV141 ) @ ( cross_product @ SV96 @ SY310 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[386]) ).
thf(458,plain,
! [SV142: $i,SV97: $i,SV15: $i] :
( ( ~ ( member @ SV15 @ ( cross_product @ SV97 @ SV142 ) )
| ( ( ordered_pair @ ( first @ SV15 ) @ ( second @ SV15 ) )
= SV15 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[387]) ).
thf(459,plain,
! [SV98: $i,SV16: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV16 @ SV98 ) @ element_relation ) )
= $true )
| ( ( member @ SV16 @ SV98 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[388]) ).
thf(460,plain,
! [SV99: $i,SV17: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV17 @ SV99 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( ~ ( member @ SV17 @ SV99 )
| ( member @ ( ordered_pair @ SV17 @ SV99 ) @ element_relation ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[389]) ).
thf(461,plain,
! [SV143: $i,SV100: $i,SV18: $i] :
( ( ~ ( member @ SV18 @ ( intersection @ SV100 @ SV143 ) )
| ( member @ SV18 @ SV100 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[390]) ).
thf(462,plain,
! [SV144: $i,SV101: $i,SV19: $i] :
( ( ~ ( member @ SV19 @ ( intersection @ SV101 @ SV144 ) )
| ( member @ SV19 @ SV144 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[391]) ).
thf(463,plain,
! [SV145: $i,SV102: $i,SV20: $i] :
( ( ~ ( member @ SV20 @ SV102 )
| ~ ( member @ SV20 @ SV145 )
| ( member @ SV20 @ ( intersection @ SV102 @ SV145 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[392]) ).
thf(464,plain,
! [SV103: $i,SV21: $i] :
( ( ( ~ ( member @ SV21 @ ( complement @ SV103 ) ) )
= $true )
| ( ( ~ ( member @ SV21 @ SV103 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[393]) ).
thf(465,plain,
! [SV104: $i,SV22: $i] :
( ( ( ~ ( member @ SV22 @ universal_class ) )
= $true )
| ( ( ( member @ SV22 @ ( complement @ SV104 ) )
| ( member @ SV22 @ SV104 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[394]) ).
thf(466,plain,
! [SV146: $i,SV107: $i,SV25: $i] :
( ( ( intersection @ SV25 @ ( cross_product @ SV107 @ SV146 ) )
= ( restrict @ SV25 @ SV107 @ SV146 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[397]) ).
thf(467,plain,
! [SV147: $i,SV108: $i,SV26: $i] :
( ( ( intersection @ ( cross_product @ SV26 @ SV108 ) @ SV147 )
= ( restrict @ SV147 @ SV26 @ SV108 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[398]) ).
thf(468,plain,
! [SV109: $i,SV27: $i] :
( ( ( ~ ( member @ SV27 @ universal_class ) )
= $true )
| ( ( ( ( restrict @ SV109 @ ( singleton @ SV27 ) @ universal_class )
= null_class )
| ( member @ SV27 @ ( domain_of @ SV109 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[399]) ).
thf(469,plain,
! [SV148: $i,SV110: $i,SV29: $i] :
( ( ! [SY311: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV29 @ SV110 ) @ SV148 ) @ ( rotate @ SY311 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV110 @ SV148 ) @ SV29 ) @ SY311 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[400]) ).
thf(470,plain,
! [SV149: $i,SV111: $i,SV30: $i] :
( ( ! [SY312: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV30 @ SV111 ) @ SV149 ) @ SY312 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( rotate @ SY312 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[401]) ).
thf(471,plain,
! [SV150: $i,SV112: $i,SV32: $i] :
( ( ! [SY313: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV112 ) @ SV150 ) @ ( flip @ SY313 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV112 @ SV32 ) @ SV150 ) @ SY313 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[402]) ).
thf(472,plain,
! [SV151: $i,SV113: $i,SV33: $i] :
( ( ! [SY314: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV113 ) @ SV151 ) @ SY314 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( flip @ SY314 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[403]) ).
thf(473,plain,
! [SV152: $i,SV114: $i,SV36: $i] :
( ( ( first @ ( not_subclass_element @ ( restrict @ SV36 @ SV114 @ ( singleton @ SV152 ) ) @ null_class ) )
= ( domain @ SV36 @ SV114 @ SV152 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[404]) ).
thf(474,plain,
! [SV153: $i,SV115: $i,SV37: $i] :
( ( ( second @ ( not_subclass_element @ ( restrict @ SV37 @ ( singleton @ SV115 ) @ SV153 ) @ null_class ) )
= ( range @ SV37 @ SV115 @ SV153 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[405]) ).
thf(475,plain,
! [SV117: $i,SV40: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV40 @ SV117 ) @ successor_relation ) )
= $true )
| ( ( ( successor @ SV40 )
= SV117 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[407]) ).
thf(476,plain,
! [SV41: $i] :
( ( ( inductive @ SV41 )
= $false )
| ( ( member @ null_class @ SV41 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[408]) ).
thf(477,plain,
! [SV42: $i] :
( ( ( inductive @ SV42 )
= $false )
| ( ( subclass @ ( image @ successor_relation @ SV42 ) @ SV42 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[409]) ).
thf(478,plain,
! [SV43: $i] :
( ( ( member @ null_class @ SV43 )
= $false )
| ( ( ~ ( subclass @ ( image @ successor_relation @ SV43 ) @ SV43 )
| ( inductive @ SV43 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[410]) ).
thf(479,plain,
! [SV44: $i] :
( ( ( inductive @ SV44 )
= $false )
| ( ( subclass @ omega @ SV44 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[411]) ).
thf(480,plain,
! [SV46: $i] :
( ( ( member @ SV46 @ universal_class )
= $false )
| ( ( member @ ( sum_class @ SV46 ) @ universal_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[412]) ).
thf(481,plain,
! [SV48: $i] :
( ( ( member @ SV48 @ universal_class )
= $false )
| ( ( member @ ( power_class @ SV48 ) @ universal_class )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[413]) ).
thf(482,plain,
! [SV154: $i,SV119: $i,SV50: $i] :
( ( ! [SY315: $i] :
( ~ ( member @ ( ordered_pair @ SV50 @ SV119 ) @ ( compose @ SV154 @ SY315 ) )
| ( member @ SV119 @ ( image @ SV154 @ ( image @ SY315 @ ( singleton @ SV50 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[415]) ).
thf(483,plain,
! [SV155: $i,SV120: $i,SV51: $i] :
( ( ! [SY316: $i] :
( ~ ( member @ SV51 @ ( image @ SV120 @ ( image @ SV155 @ ( singleton @ SY316 ) ) ) )
| ~ ( member @ ( ordered_pair @ SY316 @ SV51 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SY316 @ SV51 ) @ ( compose @ SV120 @ SV155 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[416]) ).
thf(484,plain,
! [SV52: $i] :
( ( ( single_valued_class @ SV52 )
= $false )
| ( ( subclass @ ( compose @ SV52 @ ( inverse @ SV52 ) ) @ identity_relation )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[417]) ).
thf(485,plain,
! [SV53: $i] :
( ( ( subclass @ ( compose @ SV53 @ ( inverse @ SV53 ) ) @ identity_relation )
= $false )
| ( ( single_valued_class @ SV53 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[418]) ).
thf(486,plain,
! [SV54: $i] :
( ( ( function @ SV54 )
= $false )
| ( ( subclass @ SV54 @ ( cross_product @ universal_class @ universal_class ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[419]) ).
thf(487,plain,
! [SV55: $i] :
( ( ( function @ SV55 )
= $false )
| ( ( subclass @ ( compose @ SV55 @ ( inverse @ SV55 ) ) @ identity_relation )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[420]) ).
thf(488,plain,
! [SV56: $i] :
( ( ( subclass @ SV56 @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( ~ ( subclass @ ( compose @ SV56 @ ( inverse @ SV56 ) ) @ identity_relation )
| ( function @ SV56 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[421]) ).
thf(489,plain,
! [SV121: $i,SV57: $i] :
( ( ( ~ ( function @ SV57 ) )
= $true )
| ( ( ~ ( member @ SV121 @ universal_class )
| ( member @ ( image @ SV57 @ SV121 ) @ universal_class ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[422]) ).
thf(490,plain,
! [SV61: $i] :
( ( ( member @ SV61 @ universal_class )
= $false )
| ( ( ( SV61 = null_class )
| ( member @ ( apply @ choice @ SV61 ) @ SV61 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[426]) ).
thf(491,plain,
! [SV62: $i] :
( ( ( one_to_one @ SV62 )
= $false )
| ( ( function @ SV62 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[427]) ).
thf(492,plain,
! [SV63: $i] :
( ( ( one_to_one @ SV63 )
= $false )
| ( ( function @ ( inverse @ SV63 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[428]) ).
thf(493,plain,
! [SV64: $i] :
( ( ( function @ ( inverse @ SV64 ) )
= $false )
| ( ( ~ ( function @ SV64 )
| ( one_to_one @ SV64 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[429]) ).
thf(494,plain,
! [SV67: $i] :
( ( ( operation @ SV67 )
= $false )
| ( ( function @ SV67 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[430]) ).
thf(495,plain,
! [SV68: $i] :
( ( ( operation @ SV68 )
= $false )
| ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV68 ) ) @ ( domain_of @ ( domain_of @ SV68 ) ) )
= ( domain_of @ SV68 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[431]) ).
thf(496,plain,
! [SV69: $i] :
( ( ( operation @ SV69 )
= $false )
| ( ( subclass @ ( range_of @ SV69 ) @ ( domain_of @ ( domain_of @ SV69 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[432]) ).
thf(497,plain,
! [SV156: $i,SV123: $i,SV70: $i] :
( ( ~ ( compatible @ SV70 @ SV123 @ SV156 )
| ( function @ SV70 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[433]) ).
thf(498,plain,
! [SV157: $i,SV124: $i,SV71: $i] :
( ( ~ ( compatible @ SV71 @ SV124 @ SV157 )
| ( ( domain_of @ ( domain_of @ SV124 ) )
= ( domain_of @ SV71 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[434]) ).
thf(499,plain,
! [SV158: $i,SV125: $i,SV72: $i] :
( ( ~ ( compatible @ SV72 @ SV125 @ SV158 )
| ( subclass @ ( range_of @ SV72 ) @ ( domain_of @ ( domain_of @ SV158 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[435]) ).
thf(500,plain,
! [SV159: $i,SV126: $i,SV73: $i] :
( ( ~ ( homomorphism @ SV73 @ SV126 @ SV159 )
| ( operation @ SV126 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[436]) ).
thf(501,plain,
! [SV160: $i,SV127: $i,SV74: $i] :
( ( ~ ( homomorphism @ SV74 @ SV127 @ SV160 )
| ( operation @ SV160 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[437]) ).
thf(502,plain,
! [SV161: $i,SV128: $i,SV75: $i] :
( ( ~ ( homomorphism @ SV75 @ SV128 @ SV161 )
| ( compatible @ SV75 @ SV128 @ SV161 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[438]) ).
thf(503,plain,
! [SV162: $i,SV129: $i,SV76: $i] :
( ( ! [SY317: $i,SY318: $i] :
( ~ ( homomorphism @ SV76 @ SV129 @ SV162 )
| ~ ( member @ ( ordered_pair @ SY317 @ SY318 ) @ ( domain_of @ SV129 ) )
| ( ( apply @ SV162 @ ( ordered_pair @ ( apply @ SV76 @ SY317 ) @ ( apply @ SV76 @ SY318 ) ) )
= ( apply @ SV76 @ ( apply @ SV129 @ ( ordered_pair @ SY317 @ SY318 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[439]) ).
thf(504,plain,
! [SV163: $i,SV130: $i,SV77: $i] :
( ( ~ ( operation @ SV77 )
| ~ ( operation @ SV130 )
| ~ ( compatible @ SV163 @ SV77 @ SV130 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV163 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SV163 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SV163 @ SV77 @ SV130 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[440]) ).
thf(505,plain,
! [SV164: $i,SV131: $i,SV78: $i] :
( ( ~ ( function @ SV78 )
| ( ( domain_of @ ( domain_of @ SV131 ) )
!= ( domain_of @ SV78 ) )
| ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV164 ) ) )
| ( compatible @ SV78 @ SV131 @ SV164 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[441]) ).
thf(506,plain,
! [SV132: $i,SV79: $i] :
( ( ( SV79 != SV132 )
= $true )
| ( ( subclass @ SV79 @ SV132 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[442]) ).
thf(507,plain,
! [SV133: $i,SV80: $i] :
( ( ( SV80 != SV133 )
= $true )
| ( ( subclass @ SV133 @ SV80 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[443]) ).
thf(508,plain,
! [SV81: $i] :
( ( ( function @ SV81 )
= $false )
| ( ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV81 ) ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
!= ( domain_of @ SV81 ) )
| ~ ( subclass @ ( range_of @ SV81 ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
| ( operation @ SV81 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[444]) ).
thf(509,plain,
! [SV165: $i,SV134: $i,SV82: $i] :
( ( ~ ( operation @ SV82 )
| ~ ( operation @ SV134 )
| ~ ( compatible @ SV165 @ SV82 @ SV134 )
| ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
| ( homomorphism @ SV165 @ SV82 @ SV134 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[445]) ).
thf(510,plain,
! [SV135: $i,SV83: $i] :
( ( ( ( successor @ SV83 )
!= SV135 )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ SV83 @ SV135 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV83 @ SV135 ) @ successor_relation ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[446]) ).
thf(511,plain,
! [SV136: $i,SV84: $i] :
( ( ( ( restrict @ SV84 @ ( singleton @ SV136 ) @ universal_class )
!= null_class )
= $true )
| ( ( ~ ( member @ SV136 @ ( domain_of @ SV84 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[447]) ).
thf(512,plain,
! [SV137: $i,SV85: $i,SV1: $i] :
( ( ( ~ ( subclass @ SV1 @ SV85 ) )
= $true )
| ( ( ~ ( member @ SV137 @ SV1 )
| ( member @ SV137 @ SV85 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[448]) ).
thf(513,plain,
! [SV87: $i,SV3: $i] :
( ( ( member @ ( not_subclass_element @ SV3 @ SV87 ) @ SV87 )
= $false )
| ( ( subclass @ SV3 @ SV87 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[450]) ).
thf(514,plain,
! [SV88: $i,SV5: $i] :
( ( ( subclass @ SV5 @ SV88 )
= $false )
| ( ( ~ ( subclass @ SV88 @ SV5 )
| ( SV5 = SV88 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[451]) ).
thf(515,plain,
! [SV138: $i,SV89: $i,SV6: $i] :
( ( ( ~ ( member @ SV6 @ ( unordered_pair @ SV89 @ SV138 ) ) )
= $true )
| ( ( ( SV6 = SV89 )
| ( SV6 = SV138 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[452]) ).
thf(516,plain,
! [SV90: $i,SV7: $i] :
( ( ( member @ SV7 @ universal_class )
= $false )
| ( ( member @ SV7 @ ( unordered_pair @ SV7 @ SV90 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[453]) ).
thf(517,plain,
! [SV91: $i,SV8: $i] :
( ( ( member @ SV8 @ universal_class )
= $false )
| ( ( member @ SV8 @ ( unordered_pair @ SV91 @ SV8 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[454]) ).
thf(518,plain,
! [SV166: $i,SV139: $i,SV94: $i,SV12: $i] :
( ( ~ ( member @ ( ordered_pair @ SV12 @ SV94 ) @ ( cross_product @ SV139 @ SV166 ) )
| ( member @ SV12 @ SV139 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[455]) ).
thf(519,plain,
! [SV167: $i,SV140: $i,SV95: $i,SV13: $i] :
( ( ~ ( member @ ( ordered_pair @ SV13 @ SV95 ) @ ( cross_product @ SV140 @ SV167 ) )
| ( member @ SV95 @ SV167 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[456]) ).
thf(520,plain,
! [SV168: $i,SV141: $i,SV96: $i,SV14: $i] :
( ( ~ ( member @ SV14 @ SV96 )
| ~ ( member @ SV141 @ SV168 )
| ( member @ ( ordered_pair @ SV14 @ SV141 ) @ ( cross_product @ SV96 @ SV168 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[457]) ).
thf(521,plain,
! [SV142: $i,SV97: $i,SV15: $i] :
( ( ( ~ ( member @ SV15 @ ( cross_product @ SV97 @ SV142 ) ) )
= $true )
| ( ( ( ordered_pair @ ( first @ SV15 ) @ ( second @ SV15 ) )
= SV15 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[458]) ).
thf(522,plain,
! [SV98: $i,SV16: $i] :
( ( ( member @ ( ordered_pair @ SV16 @ SV98 ) @ element_relation )
= $false )
| ( ( member @ SV16 @ SV98 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[459]) ).
thf(523,plain,
! [SV99: $i,SV17: $i] :
( ( ( member @ ( ordered_pair @ SV17 @ SV99 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( ~ ( member @ SV17 @ SV99 )
| ( member @ ( ordered_pair @ SV17 @ SV99 ) @ element_relation ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[460]) ).
thf(524,plain,
! [SV143: $i,SV100: $i,SV18: $i] :
( ( ( ~ ( member @ SV18 @ ( intersection @ SV100 @ SV143 ) ) )
= $true )
| ( ( member @ SV18 @ SV100 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[461]) ).
thf(525,plain,
! [SV144: $i,SV101: $i,SV19: $i] :
( ( ( ~ ( member @ SV19 @ ( intersection @ SV101 @ SV144 ) ) )
= $true )
| ( ( member @ SV19 @ SV144 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[462]) ).
thf(526,plain,
! [SV145: $i,SV102: $i,SV20: $i] :
( ( ( ~ ( member @ SV20 @ SV102 ) )
= $true )
| ( ( ~ ( member @ SV20 @ SV145 )
| ( member @ SV20 @ ( intersection @ SV102 @ SV145 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[463]) ).
thf(527,plain,
! [SV103: $i,SV21: $i] :
( ( ( member @ SV21 @ ( complement @ SV103 ) )
= $false )
| ( ( ~ ( member @ SV21 @ SV103 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[464]) ).
thf(528,plain,
! [SV104: $i,SV22: $i] :
( ( ( member @ SV22 @ universal_class )
= $false )
| ( ( ( member @ SV22 @ ( complement @ SV104 ) )
| ( member @ SV22 @ SV104 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[465]) ).
thf(529,plain,
! [SV109: $i,SV27: $i] :
( ( ( member @ SV27 @ universal_class )
= $false )
| ( ( ( ( restrict @ SV109 @ ( singleton @ SV27 ) @ universal_class )
= null_class )
| ( member @ SV27 @ ( domain_of @ SV109 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[468]) ).
thf(530,plain,
! [SV169: $i,SV148: $i,SV110: $i,SV29: $i] :
( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV29 @ SV110 ) @ SV148 ) @ ( rotate @ SV169 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV110 @ SV148 ) @ SV29 ) @ SV169 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[469]) ).
thf(531,plain,
! [SV170: $i,SV149: $i,SV111: $i,SV30: $i] :
( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV30 @ SV111 ) @ SV149 ) @ SV170 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( rotate @ SV170 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[470]) ).
thf(532,plain,
! [SV171: $i,SV150: $i,SV112: $i,SV32: $i] :
( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV112 ) @ SV150 ) @ ( flip @ SV171 ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV112 @ SV32 ) @ SV150 ) @ SV171 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[471]) ).
thf(533,plain,
! [SV172: $i,SV151: $i,SV113: $i,SV33: $i] :
( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV113 ) @ SV151 ) @ SV172 )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( flip @ SV172 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[472]) ).
thf(534,plain,
! [SV117: $i,SV40: $i] :
( ( ( member @ ( ordered_pair @ SV40 @ SV117 ) @ successor_relation )
= $false )
| ( ( ( successor @ SV40 )
= SV117 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[475]) ).
thf(535,plain,
! [SV43: $i] :
( ( ( ~ ( subclass @ ( image @ successor_relation @ SV43 ) @ SV43 ) )
= $true )
| ( ( inductive @ SV43 )
= $true )
| ( ( member @ null_class @ SV43 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[478]) ).
thf(536,plain,
! [SV173: $i,SV154: $i,SV119: $i,SV50: $i] :
( ( ~ ( member @ ( ordered_pair @ SV50 @ SV119 ) @ ( compose @ SV154 @ SV173 ) )
| ( member @ SV119 @ ( image @ SV154 @ ( image @ SV173 @ ( singleton @ SV50 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[482]) ).
thf(537,plain,
! [SV174: $i,SV155: $i,SV120: $i,SV51: $i] :
( ( ~ ( member @ SV51 @ ( image @ SV120 @ ( image @ SV155 @ ( singleton @ SV174 ) ) ) )
| ~ ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( compose @ SV120 @ SV155 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[483]) ).
thf(538,plain,
! [SV56: $i] :
( ( ( ~ ( subclass @ ( compose @ SV56 @ ( inverse @ SV56 ) ) @ identity_relation ) )
= $true )
| ( ( function @ SV56 )
= $true )
| ( ( subclass @ SV56 @ ( cross_product @ universal_class @ universal_class ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[488]) ).
thf(539,plain,
! [SV121: $i,SV57: $i] :
( ( ( function @ SV57 )
= $false )
| ( ( ~ ( member @ SV121 @ universal_class )
| ( member @ ( image @ SV57 @ SV121 ) @ universal_class ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[489]) ).
thf(540,plain,
! [SV61: $i] :
( ( ( SV61 = null_class )
= $true )
| ( ( member @ ( apply @ choice @ SV61 ) @ SV61 )
= $true )
| ( ( member @ SV61 @ universal_class )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[490]) ).
thf(541,plain,
! [SV64: $i] :
( ( ( ~ ( function @ SV64 ) )
= $true )
| ( ( one_to_one @ SV64 )
= $true )
| ( ( function @ ( inverse @ SV64 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[493]) ).
thf(542,plain,
! [SV156: $i,SV123: $i,SV70: $i] :
( ( ( ~ ( compatible @ SV70 @ SV123 @ SV156 ) )
= $true )
| ( ( function @ SV70 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[497]) ).
thf(543,plain,
! [SV157: $i,SV124: $i,SV71: $i] :
( ( ( ~ ( compatible @ SV71 @ SV124 @ SV157 ) )
= $true )
| ( ( ( domain_of @ ( domain_of @ SV124 ) )
= ( domain_of @ SV71 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[498]) ).
thf(544,plain,
! [SV158: $i,SV125: $i,SV72: $i] :
( ( ( ~ ( compatible @ SV72 @ SV125 @ SV158 ) )
= $true )
| ( ( subclass @ ( range_of @ SV72 ) @ ( domain_of @ ( domain_of @ SV158 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[499]) ).
thf(545,plain,
! [SV159: $i,SV126: $i,SV73: $i] :
( ( ( ~ ( homomorphism @ SV73 @ SV126 @ SV159 ) )
= $true )
| ( ( operation @ SV126 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[500]) ).
thf(546,plain,
! [SV160: $i,SV127: $i,SV74: $i] :
( ( ( ~ ( homomorphism @ SV74 @ SV127 @ SV160 ) )
= $true )
| ( ( operation @ SV160 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[501]) ).
thf(547,plain,
! [SV161: $i,SV128: $i,SV75: $i] :
( ( ( ~ ( homomorphism @ SV75 @ SV128 @ SV161 ) )
= $true )
| ( ( compatible @ SV75 @ SV128 @ SV161 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[502]) ).
thf(548,plain,
! [SV175: $i,SV162: $i,SV129: $i,SV76: $i] :
( ( ! [SY319: $i] :
( ~ ( homomorphism @ SV76 @ SV129 @ SV162 )
| ~ ( member @ ( ordered_pair @ SV175 @ SY319 ) @ ( domain_of @ SV129 ) )
| ( ( apply @ SV162 @ ( ordered_pair @ ( apply @ SV76 @ SV175 ) @ ( apply @ SV76 @ SY319 ) ) )
= ( apply @ SV76 @ ( apply @ SV129 @ ( ordered_pair @ SV175 @ SY319 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[503]) ).
thf(549,plain,
! [SV163: $i,SV130: $i,SV77: $i] :
( ( ( ~ ( operation @ SV77 ) )
= $true )
| ( ( ~ ( operation @ SV130 )
| ~ ( compatible @ SV163 @ SV77 @ SV130 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV163 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SV163 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SV163 @ SV77 @ SV130 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[504]) ).
thf(550,plain,
! [SV164: $i,SV131: $i,SV78: $i] :
( ( ( ~ ( function @ SV78 ) )
= $true )
| ( ( ( ( domain_of @ ( domain_of @ SV131 ) )
!= ( domain_of @ SV78 ) )
| ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV164 ) ) )
| ( compatible @ SV78 @ SV131 @ SV164 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[505]) ).
thf(551,plain,
! [SV132: $i,SV79: $i] :
( ( ( SV79 = SV132 )
= $false )
| ( ( subclass @ SV79 @ SV132 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[506]) ).
thf(552,plain,
! [SV133: $i,SV80: $i] :
( ( ( SV80 = SV133 )
= $false )
| ( ( subclass @ SV133 @ SV80 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[507]) ).
thf(553,plain,
! [SV81: $i] :
( ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV81 ) ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
!= ( domain_of @ SV81 ) )
= $true )
| ( ( ~ ( subclass @ ( range_of @ SV81 ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
| ( operation @ SV81 ) )
= $true )
| ( ( function @ SV81 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[508]) ).
thf(554,plain,
! [SV165: $i,SV134: $i,SV82: $i] :
( ( ( ~ ( operation @ SV82 ) )
= $true )
| ( ( ~ ( operation @ SV134 )
| ~ ( compatible @ SV165 @ SV82 @ SV134 )
| ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
| ( homomorphism @ SV165 @ SV82 @ SV134 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[509]) ).
thf(555,plain,
! [SV135: $i,SV83: $i] :
( ( ( ( successor @ SV83 )
= SV135 )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ SV83 @ SV135 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV83 @ SV135 ) @ successor_relation ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[510]) ).
thf(556,plain,
! [SV136: $i,SV84: $i] :
( ( ( ( restrict @ SV84 @ ( singleton @ SV136 ) @ universal_class )
= null_class )
= $false )
| ( ( ~ ( member @ SV136 @ ( domain_of @ SV84 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[511]) ).
thf(557,plain,
! [SV137: $i,SV85: $i,SV1: $i] :
( ( ( subclass @ SV1 @ SV85 )
= $false )
| ( ( ~ ( member @ SV137 @ SV1 )
| ( member @ SV137 @ SV85 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[512]) ).
thf(558,plain,
! [SV5: $i,SV88: $i] :
( ( ( ~ ( subclass @ SV88 @ SV5 ) )
= $true )
| ( ( SV5 = SV88 )
= $true )
| ( ( subclass @ SV5 @ SV88 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[514]) ).
thf(559,plain,
! [SV138: $i,SV89: $i,SV6: $i] :
( ( ( member @ SV6 @ ( unordered_pair @ SV89 @ SV138 ) )
= $false )
| ( ( ( SV6 = SV89 )
| ( SV6 = SV138 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[515]) ).
thf(560,plain,
! [SV166: $i,SV139: $i,SV94: $i,SV12: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV12 @ SV94 ) @ ( cross_product @ SV139 @ SV166 ) ) )
= $true )
| ( ( member @ SV12 @ SV139 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[518]) ).
thf(561,plain,
! [SV167: $i,SV140: $i,SV95: $i,SV13: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV13 @ SV95 ) @ ( cross_product @ SV140 @ SV167 ) ) )
= $true )
| ( ( member @ SV95 @ SV167 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[519]) ).
thf(562,plain,
! [SV168: $i,SV141: $i,SV96: $i,SV14: $i] :
( ( ( ~ ( member @ SV14 @ SV96 ) )
= $true )
| ( ( ~ ( member @ SV141 @ SV168 )
| ( member @ ( ordered_pair @ SV14 @ SV141 ) @ ( cross_product @ SV96 @ SV168 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[520]) ).
thf(563,plain,
! [SV142: $i,SV97: $i,SV15: $i] :
( ( ( member @ SV15 @ ( cross_product @ SV97 @ SV142 ) )
= $false )
| ( ( ( ordered_pair @ ( first @ SV15 ) @ ( second @ SV15 ) )
= SV15 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[521]) ).
thf(564,plain,
! [SV99: $i,SV17: $i] :
( ( ( ~ ( member @ SV17 @ SV99 ) )
= $true )
| ( ( member @ ( ordered_pair @ SV17 @ SV99 ) @ element_relation )
= $true )
| ( ( member @ ( ordered_pair @ SV17 @ SV99 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[523]) ).
thf(565,plain,
! [SV143: $i,SV100: $i,SV18: $i] :
( ( ( member @ SV18 @ ( intersection @ SV100 @ SV143 ) )
= $false )
| ( ( member @ SV18 @ SV100 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[524]) ).
thf(566,plain,
! [SV144: $i,SV101: $i,SV19: $i] :
( ( ( member @ SV19 @ ( intersection @ SV101 @ SV144 ) )
= $false )
| ( ( member @ SV19 @ SV144 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[525]) ).
thf(567,plain,
! [SV145: $i,SV102: $i,SV20: $i] :
( ( ( member @ SV20 @ SV102 )
= $false )
| ( ( ~ ( member @ SV20 @ SV145 )
| ( member @ SV20 @ ( intersection @ SV102 @ SV145 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[526]) ).
thf(568,plain,
! [SV103: $i,SV21: $i] :
( ( ( member @ SV21 @ SV103 )
= $false )
| ( ( member @ SV21 @ ( complement @ SV103 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[527]) ).
thf(569,plain,
! [SV104: $i,SV22: $i] :
( ( ( member @ SV22 @ ( complement @ SV104 ) )
= $true )
| ( ( member @ SV22 @ SV104 )
= $true )
| ( ( member @ SV22 @ universal_class )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[528]) ).
thf(570,plain,
! [SV27: $i,SV109: $i] :
( ( ( ( restrict @ SV109 @ ( singleton @ SV27 ) @ universal_class )
= null_class )
= $true )
| ( ( member @ SV27 @ ( domain_of @ SV109 ) )
= $true )
| ( ( member @ SV27 @ universal_class )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[529]) ).
thf(571,plain,
! [SV169: $i,SV148: $i,SV110: $i,SV29: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV29 @ SV110 ) @ SV148 ) @ ( rotate @ SV169 ) ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV110 @ SV148 ) @ SV29 ) @ SV169 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[530]) ).
thf(572,plain,
! [SV170: $i,SV149: $i,SV111: $i,SV30: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV30 @ SV111 ) @ SV149 ) @ SV170 ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( rotate @ SV170 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[531]) ).
thf(573,plain,
! [SV171: $i,SV150: $i,SV112: $i,SV32: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV112 ) @ SV150 ) @ ( flip @ SV171 ) ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV112 @ SV32 ) @ SV150 ) @ SV171 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[532]) ).
thf(574,plain,
! [SV172: $i,SV151: $i,SV113: $i,SV33: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV113 ) @ SV151 ) @ SV172 ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( flip @ SV172 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[533]) ).
thf(575,plain,
! [SV43: $i] :
( ( ( subclass @ ( image @ successor_relation @ SV43 ) @ SV43 )
= $false )
| ( ( inductive @ SV43 )
= $true )
| ( ( member @ null_class @ SV43 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[535]) ).
thf(576,plain,
! [SV173: $i,SV154: $i,SV119: $i,SV50: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV50 @ SV119 ) @ ( compose @ SV154 @ SV173 ) ) )
= $true )
| ( ( member @ SV119 @ ( image @ SV154 @ ( image @ SV173 @ ( singleton @ SV50 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[536]) ).
thf(577,plain,
! [SV174: $i,SV155: $i,SV120: $i,SV51: $i] :
( ( ( ~ ( member @ SV51 @ ( image @ SV120 @ ( image @ SV155 @ ( singleton @ SV174 ) ) ) ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( compose @ SV120 @ SV155 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[537]) ).
thf(578,plain,
! [SV56: $i] :
( ( ( subclass @ ( compose @ SV56 @ ( inverse @ SV56 ) ) @ identity_relation )
= $false )
| ( ( function @ SV56 )
= $true )
| ( ( subclass @ SV56 @ ( cross_product @ universal_class @ universal_class ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[538]) ).
thf(579,plain,
! [SV57: $i,SV121: $i] :
( ( ( ~ ( member @ SV121 @ universal_class ) )
= $true )
| ( ( member @ ( image @ SV57 @ SV121 ) @ universal_class )
= $true )
| ( ( function @ SV57 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[539]) ).
thf(580,plain,
! [SV64: $i] :
( ( ( function @ SV64 )
= $false )
| ( ( one_to_one @ SV64 )
= $true )
| ( ( function @ ( inverse @ SV64 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[541]) ).
thf(581,plain,
! [SV156: $i,SV123: $i,SV70: $i] :
( ( ( compatible @ SV70 @ SV123 @ SV156 )
= $false )
| ( ( function @ SV70 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[542]) ).
thf(582,plain,
! [SV157: $i,SV124: $i,SV71: $i] :
( ( ( compatible @ SV71 @ SV124 @ SV157 )
= $false )
| ( ( ( domain_of @ ( domain_of @ SV124 ) )
= ( domain_of @ SV71 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[543]) ).
thf(583,plain,
! [SV158: $i,SV125: $i,SV72: $i] :
( ( ( compatible @ SV72 @ SV125 @ SV158 )
= $false )
| ( ( subclass @ ( range_of @ SV72 ) @ ( domain_of @ ( domain_of @ SV158 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[544]) ).
thf(584,plain,
! [SV159: $i,SV126: $i,SV73: $i] :
( ( ( homomorphism @ SV73 @ SV126 @ SV159 )
= $false )
| ( ( operation @ SV126 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[545]) ).
thf(585,plain,
! [SV160: $i,SV127: $i,SV74: $i] :
( ( ( homomorphism @ SV74 @ SV127 @ SV160 )
= $false )
| ( ( operation @ SV160 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[546]) ).
thf(586,plain,
! [SV161: $i,SV128: $i,SV75: $i] :
( ( ( homomorphism @ SV75 @ SV128 @ SV161 )
= $false )
| ( ( compatible @ SV75 @ SV128 @ SV161 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[547]) ).
thf(587,plain,
! [SV176: $i,SV175: $i,SV162: $i,SV129: $i,SV76: $i] :
( ( ~ ( homomorphism @ SV76 @ SV129 @ SV162 )
| ~ ( member @ ( ordered_pair @ SV175 @ SV176 ) @ ( domain_of @ SV129 ) )
| ( ( apply @ SV162 @ ( ordered_pair @ ( apply @ SV76 @ SV175 ) @ ( apply @ SV76 @ SV176 ) ) )
= ( apply @ SV76 @ ( apply @ SV129 @ ( ordered_pair @ SV175 @ SV176 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[548]) ).
thf(588,plain,
! [SV163: $i,SV130: $i,SV77: $i] :
( ( ( operation @ SV77 )
= $false )
| ( ( ~ ( operation @ SV130 )
| ~ ( compatible @ SV163 @ SV77 @ SV130 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV163 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SV163 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SV163 @ SV77 @ SV130 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[549]) ).
thf(589,plain,
! [SV164: $i,SV131: $i,SV78: $i] :
( ( ( function @ SV78 )
= $false )
| ( ( ( ( domain_of @ ( domain_of @ SV131 ) )
!= ( domain_of @ SV78 ) )
| ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV164 ) ) )
| ( compatible @ SV78 @ SV131 @ SV164 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[550]) ).
thf(590,plain,
! [SV81: $i] :
( ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV81 ) ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
= ( domain_of @ SV81 ) )
= $false )
| ( ( ~ ( subclass @ ( range_of @ SV81 ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
| ( operation @ SV81 ) )
= $true )
| ( ( function @ SV81 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[553]) ).
thf(591,plain,
! [SV165: $i,SV134: $i,SV82: $i] :
( ( ( operation @ SV82 )
= $false )
| ( ( ~ ( operation @ SV134 )
| ~ ( compatible @ SV165 @ SV82 @ SV134 )
| ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
| ( homomorphism @ SV165 @ SV82 @ SV134 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[554]) ).
thf(592,plain,
! [SV135: $i,SV83: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV83 @ SV135 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( member @ ( ordered_pair @ SV83 @ SV135 ) @ successor_relation )
= $true )
| ( ( ( successor @ SV83 )
= SV135 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[555]) ).
thf(593,plain,
! [SV84: $i,SV136: $i] :
( ( ( member @ SV136 @ ( domain_of @ SV84 ) )
= $false )
| ( ( ( restrict @ SV84 @ ( singleton @ SV136 ) @ universal_class )
= null_class )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[556]) ).
thf(594,plain,
! [SV85: $i,SV1: $i,SV137: $i] :
( ( ( ~ ( member @ SV137 @ SV1 ) )
= $true )
| ( ( member @ SV137 @ SV85 )
= $true )
| ( ( subclass @ SV1 @ SV85 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[557]) ).
thf(595,plain,
! [SV5: $i,SV88: $i] :
( ( ( subclass @ SV88 @ SV5 )
= $false )
| ( ( SV5 = SV88 )
= $true )
| ( ( subclass @ SV5 @ SV88 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[558]) ).
thf(596,plain,
! [SV138: $i,SV89: $i,SV6: $i] :
( ( ( SV6 = SV89 )
= $true )
| ( ( SV6 = SV138 )
= $true )
| ( ( member @ SV6 @ ( unordered_pair @ SV89 @ SV138 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[559]) ).
thf(597,plain,
! [SV166: $i,SV139: $i,SV94: $i,SV12: $i] :
( ( ( member @ ( ordered_pair @ SV12 @ SV94 ) @ ( cross_product @ SV139 @ SV166 ) )
= $false )
| ( ( member @ SV12 @ SV139 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[560]) ).
thf(598,plain,
! [SV167: $i,SV140: $i,SV95: $i,SV13: $i] :
( ( ( member @ ( ordered_pair @ SV13 @ SV95 ) @ ( cross_product @ SV140 @ SV167 ) )
= $false )
| ( ( member @ SV95 @ SV167 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[561]) ).
thf(599,plain,
! [SV168: $i,SV141: $i,SV96: $i,SV14: $i] :
( ( ( member @ SV14 @ SV96 )
= $false )
| ( ( ~ ( member @ SV141 @ SV168 )
| ( member @ ( ordered_pair @ SV14 @ SV141 ) @ ( cross_product @ SV96 @ SV168 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[562]) ).
thf(600,plain,
! [SV99: $i,SV17: $i] :
( ( ( member @ SV17 @ SV99 )
= $false )
| ( ( member @ ( ordered_pair @ SV17 @ SV99 ) @ element_relation )
= $true )
| ( ( member @ ( ordered_pair @ SV17 @ SV99 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[564]) ).
thf(601,plain,
! [SV102: $i,SV145: $i,SV20: $i] :
( ( ( ~ ( member @ SV20 @ SV145 ) )
= $true )
| ( ( member @ SV20 @ ( intersection @ SV102 @ SV145 ) )
= $true )
| ( ( member @ SV20 @ SV102 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[567]) ).
thf(602,plain,
! [SV169: $i,SV148: $i,SV110: $i,SV29: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV29 @ SV110 ) @ SV148 ) @ ( rotate @ SV169 ) )
= $false )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV110 @ SV148 ) @ SV29 ) @ SV169 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[571]) ).
thf(603,plain,
! [SV170: $i,SV149: $i,SV111: $i,SV30: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV30 @ SV111 ) @ SV149 ) @ SV170 )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( rotate @ SV170 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[572]) ).
thf(604,plain,
! [SV171: $i,SV150: $i,SV112: $i,SV32: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV32 @ SV112 ) @ SV150 ) @ ( flip @ SV171 ) )
= $false )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV112 @ SV32 ) @ SV150 ) @ SV171 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[573]) ).
thf(605,plain,
! [SV172: $i,SV151: $i,SV113: $i,SV33: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV113 ) @ SV151 ) @ SV172 )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( flip @ SV172 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[574]) ).
thf(606,plain,
! [SV173: $i,SV154: $i,SV119: $i,SV50: $i] :
( ( ( member @ ( ordered_pair @ SV50 @ SV119 ) @ ( compose @ SV154 @ SV173 ) )
= $false )
| ( ( member @ SV119 @ ( image @ SV154 @ ( image @ SV173 @ ( singleton @ SV50 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[576]) ).
thf(607,plain,
! [SV174: $i,SV155: $i,SV120: $i,SV51: $i] :
( ( ( member @ SV51 @ ( image @ SV120 @ ( image @ SV155 @ ( singleton @ SV174 ) ) ) )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( compose @ SV120 @ SV155 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[577]) ).
thf(608,plain,
! [SV57: $i,SV121: $i] :
( ( ( member @ SV121 @ universal_class )
= $false )
| ( ( member @ ( image @ SV57 @ SV121 ) @ universal_class )
= $true )
| ( ( function @ SV57 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[579]) ).
thf(609,plain,
! [SV176: $i,SV175: $i,SV162: $i,SV129: $i,SV76: $i] :
( ( ( ~ ( homomorphism @ SV76 @ SV129 @ SV162 ) )
= $true )
| ( ( ~ ( member @ ( ordered_pair @ SV175 @ SV176 ) @ ( domain_of @ SV129 ) )
| ( ( apply @ SV162 @ ( ordered_pair @ ( apply @ SV76 @ SV175 ) @ ( apply @ SV76 @ SV176 ) ) )
= ( apply @ SV76 @ ( apply @ SV129 @ ( ordered_pair @ SV175 @ SV176 ) ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[587]) ).
thf(610,plain,
! [SV77: $i,SV163: $i,SV130: $i] :
( ( ( ~ ( operation @ SV130 ) )
= $true )
| ( ( ~ ( compatible @ SV163 @ SV77 @ SV130 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV163 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SV163 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SV163 @ SV77 @ SV130 ) )
= $true )
| ( ( operation @ SV77 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[588]) ).
thf(611,plain,
! [SV164: $i,SV78: $i,SV131: $i] :
( ( ( ( domain_of @ ( domain_of @ SV131 ) )
!= ( domain_of @ SV78 ) )
= $true )
| ( ( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV164 ) ) )
| ( compatible @ SV78 @ SV131 @ SV164 ) )
= $true )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[589]) ).
thf(612,plain,
! [SV81: $i] :
( ( ( ~ ( subclass @ ( range_of @ SV81 ) @ ( domain_of @ ( domain_of @ SV81 ) ) ) )
= $true )
| ( ( operation @ SV81 )
= $true )
| ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV81 ) ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
= ( domain_of @ SV81 ) )
= $false )
| ( ( function @ SV81 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[590]) ).
thf(613,plain,
! [SV82: $i,SV165: $i,SV134: $i] :
( ( ( ~ ( operation @ SV134 ) )
= $true )
| ( ( ~ ( compatible @ SV165 @ SV82 @ SV134 )
| ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
| ( homomorphism @ SV165 @ SV82 @ SV134 ) )
= $true )
| ( ( operation @ SV82 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[591]) ).
thf(614,plain,
! [SV135: $i,SV83: $i] :
( ( ( member @ ( ordered_pair @ SV83 @ SV135 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( member @ ( ordered_pair @ SV83 @ SV135 ) @ successor_relation )
= $true )
| ( ( ( successor @ SV83 )
= SV135 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[592]) ).
thf(615,plain,
! [SV85: $i,SV1: $i,SV137: $i] :
( ( ( member @ SV137 @ SV1 )
= $false )
| ( ( member @ SV137 @ SV85 )
= $true )
| ( ( subclass @ SV1 @ SV85 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[594]) ).
thf(616,plain,
! [SV96: $i,SV14: $i,SV168: $i,SV141: $i] :
( ( ( ~ ( member @ SV141 @ SV168 ) )
= $true )
| ( ( member @ ( ordered_pair @ SV14 @ SV141 ) @ ( cross_product @ SV96 @ SV168 ) )
= $true )
| ( ( member @ SV14 @ SV96 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[599]) ).
thf(617,plain,
! [SV102: $i,SV145: $i,SV20: $i] :
( ( ( member @ SV20 @ SV145 )
= $false )
| ( ( member @ SV20 @ ( intersection @ SV102 @ SV145 ) )
= $true )
| ( ( member @ SV20 @ SV102 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[601]) ).
thf(618,plain,
! [SV170: $i,SV111: $i,SV30: $i,SV149: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( rotate @ SV170 ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV30 @ SV111 ) @ SV149 ) @ SV170 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[603]) ).
thf(619,plain,
! [SV172: $i,SV151: $i,SV33: $i,SV113: $i] :
( ( ( ~ ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( flip @ SV172 ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV113 ) @ SV151 ) @ SV172 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[605]) ).
thf(620,plain,
! [SV155: $i,SV120: $i,SV51: $i,SV174: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( cross_product @ universal_class @ universal_class ) ) )
= $true )
| ( ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( compose @ SV120 @ SV155 ) )
= $true )
| ( ( member @ SV51 @ ( image @ SV120 @ ( image @ SV155 @ ( singleton @ SV174 ) ) ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[607]) ).
thf(621,plain,
! [SV176: $i,SV175: $i,SV162: $i,SV129: $i,SV76: $i] :
( ( ( homomorphism @ SV76 @ SV129 @ SV162 )
= $false )
| ( ( ~ ( member @ ( ordered_pair @ SV175 @ SV176 ) @ ( domain_of @ SV129 ) )
| ( ( apply @ SV162 @ ( ordered_pair @ ( apply @ SV76 @ SV175 ) @ ( apply @ SV76 @ SV176 ) ) )
= ( apply @ SV76 @ ( apply @ SV129 @ ( ordered_pair @ SV175 @ SV176 ) ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[609]) ).
thf(622,plain,
! [SV77: $i,SV163: $i,SV130: $i] :
( ( ( operation @ SV130 )
= $false )
| ( ( ~ ( compatible @ SV163 @ SV77 @ SV130 )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV163 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SV163 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SV163 @ SV77 @ SV130 ) )
= $true )
| ( ( operation @ SV77 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[610]) ).
thf(623,plain,
! [SV164: $i,SV78: $i,SV131: $i] :
( ( ( ( domain_of @ ( domain_of @ SV131 ) )
= ( domain_of @ SV78 ) )
= $false )
| ( ( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV164 ) ) )
| ( compatible @ SV78 @ SV131 @ SV164 ) )
= $true )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[611]) ).
thf(624,plain,
! [SV81: $i] :
( ( ( subclass @ ( range_of @ SV81 ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
= $false )
| ( ( operation @ SV81 )
= $true )
| ( ( ( cross_product @ ( domain_of @ ( domain_of @ SV81 ) ) @ ( domain_of @ ( domain_of @ SV81 ) ) )
= ( domain_of @ SV81 ) )
= $false )
| ( ( function @ SV81 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[612]) ).
thf(625,plain,
! [SV82: $i,SV165: $i,SV134: $i] :
( ( ( operation @ SV134 )
= $false )
| ( ( ~ ( compatible @ SV165 @ SV82 @ SV134 )
| ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
| ( homomorphism @ SV165 @ SV82 @ SV134 ) )
= $true )
| ( ( operation @ SV82 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[613]) ).
thf(626,plain,
! [SV96: $i,SV14: $i,SV168: $i,SV141: $i] :
( ( ( member @ SV141 @ SV168 )
= $false )
| ( ( member @ ( ordered_pair @ SV14 @ SV141 ) @ ( cross_product @ SV96 @ SV168 ) )
= $true )
| ( ( member @ SV14 @ SV96 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[616]) ).
thf(627,plain,
! [SV170: $i,SV111: $i,SV30: $i,SV149: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
= $false )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV149 @ SV30 ) @ SV111 ) @ ( rotate @ SV170 ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV30 @ SV111 ) @ SV149 ) @ SV170 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[618]) ).
thf(628,plain,
! [SV172: $i,SV151: $i,SV33: $i,SV113: $i] :
( ( ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
= $false )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV113 @ SV33 ) @ SV151 ) @ ( flip @ SV172 ) )
= $true )
| ( ( member @ ( ordered_pair @ ( ordered_pair @ SV33 @ SV113 ) @ SV151 ) @ SV172 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[619]) ).
thf(629,plain,
! [SV155: $i,SV120: $i,SV51: $i,SV174: $i] :
( ( ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( cross_product @ universal_class @ universal_class ) )
= $false )
| ( ( member @ ( ordered_pair @ SV174 @ SV51 ) @ ( compose @ SV120 @ SV155 ) )
= $true )
| ( ( member @ SV51 @ ( image @ SV120 @ ( image @ SV155 @ ( singleton @ SV174 ) ) ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[620]) ).
thf(630,plain,
! [SV76: $i,SV162: $i,SV129: $i,SV176: $i,SV175: $i] :
( ( ( ~ ( member @ ( ordered_pair @ SV175 @ SV176 ) @ ( domain_of @ SV129 ) ) )
= $true )
| ( ( ( apply @ SV162 @ ( ordered_pair @ ( apply @ SV76 @ SV175 ) @ ( apply @ SV76 @ SV176 ) ) )
= ( apply @ SV76 @ ( apply @ SV129 @ ( ordered_pair @ SV175 @ SV176 ) ) ) )
= $true )
| ( ( homomorphism @ SV76 @ SV129 @ SV162 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[621]) ).
thf(631,plain,
! [SV130: $i,SV77: $i,SV163: $i] :
( ( ( ~ ( compatible @ SV163 @ SV77 @ SV130 ) )
= $true )
| ( ( ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV163 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SV163 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SV163 @ SV77 @ SV130 ) )
= $true )
| ( ( operation @ SV130 )
= $false )
| ( ( operation @ SV77 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[622]) ).
thf(632,plain,
! [SV131: $i,SV164: $i,SV78: $i] :
( ( ( ~ ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV164 ) ) ) )
= $true )
| ( ( compatible @ SV78 @ SV131 @ SV164 )
= $true )
| ( ( ( domain_of @ ( domain_of @ SV131 ) )
= ( domain_of @ SV78 ) )
= $false )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[623]) ).
thf(633,plain,
! [SV134: $i,SV82: $i,SV165: $i] :
( ( ( ~ ( compatible @ SV165 @ SV82 @ SV134 ) )
= $true )
| ( ( ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
| ( homomorphism @ SV165 @ SV82 @ SV134 ) )
= $true )
| ( ( operation @ SV134 )
= $false )
| ( ( operation @ SV82 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[625]) ).
thf(634,plain,
! [SV76: $i,SV162: $i,SV129: $i,SV176: $i,SV175: $i] :
( ( ( member @ ( ordered_pair @ SV175 @ SV176 ) @ ( domain_of @ SV129 ) )
= $false )
| ( ( ( apply @ SV162 @ ( ordered_pair @ ( apply @ SV76 @ SV175 ) @ ( apply @ SV76 @ SV176 ) ) )
= ( apply @ SV76 @ ( apply @ SV129 @ ( ordered_pair @ SV175 @ SV176 ) ) ) )
= $true )
| ( ( homomorphism @ SV76 @ SV129 @ SV162 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[630]) ).
thf(635,plain,
! [SV130: $i,SV77: $i,SV163: $i] :
( ( ( compatible @ SV163 @ SV77 @ SV130 )
= $false )
| ( ( ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV163 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SV163 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
| ( homomorphism @ SV163 @ SV77 @ SV130 ) )
= $true )
| ( ( operation @ SV130 )
= $false )
| ( ( operation @ SV77 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[631]) ).
thf(636,plain,
! [SV131: $i,SV164: $i,SV78: $i] :
( ( ( subclass @ ( range_of @ SV78 ) @ ( domain_of @ ( domain_of @ SV164 ) ) )
= $false )
| ( ( compatible @ SV78 @ SV131 @ SV164 )
= $true )
| ( ( ( domain_of @ ( domain_of @ SV131 ) )
= ( domain_of @ SV78 ) )
= $false )
| ( ( function @ SV78 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[632]) ).
thf(637,plain,
! [SV134: $i,SV82: $i,SV165: $i] :
( ( ( compatible @ SV165 @ SV82 @ SV134 )
= $false )
| ( ( ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
| ( homomorphism @ SV165 @ SV82 @ SV134 ) )
= $true )
| ( ( operation @ SV134 )
= $false )
| ( ( operation @ SV82 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[633]) ).
thf(638,plain,
! [SV130: $i,SV77: $i,SV163: $i] :
( ( ( member @ ( ordered_pair @ ( not_homomorphism1 @ SV163 @ SV77 @ SV130 ) @ ( not_homomorphism2 @ SV163 @ SV77 @ SV130 ) ) @ ( domain_of @ SV77 ) )
= $true )
| ( ( homomorphism @ SV163 @ SV77 @ SV130 )
= $true )
| ( ( compatible @ SV163 @ SV77 @ SV130 )
= $false )
| ( ( operation @ SV130 )
= $false )
| ( ( operation @ SV77 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[635]) ).
thf(639,plain,
! [SV82: $i,SV165: $i,SV134: $i] :
( ( ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
!= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
= $true )
| ( ( homomorphism @ SV165 @ SV82 @ SV134 )
= $true )
| ( ( compatible @ SV165 @ SV82 @ SV134 )
= $false )
| ( ( operation @ SV134 )
= $false )
| ( ( operation @ SV82 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[637]) ).
thf(640,plain,
! [SV82: $i,SV165: $i,SV134: $i] :
( ( ( ( apply @ SV134 @ ( ordered_pair @ ( apply @ SV165 @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) ) @ ( apply @ SV165 @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) )
= ( apply @ SV165 @ ( apply @ SV82 @ ( ordered_pair @ ( not_homomorphism1 @ SV165 @ SV82 @ SV134 ) @ ( not_homomorphism2 @ SV165 @ SV82 @ SV134 ) ) ) ) )
= $false )
| ( ( homomorphism @ SV165 @ SV82 @ SV134 )
= $true )
| ( ( compatible @ SV165 @ SV82 @ SV134 )
= $false )
| ( ( operation @ SV134 )
= $false )
| ( ( operation @ SV82 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[639]) ).
thf(641,plain,
$false = $true,
inference(fo_atp_e,[status(thm)],[207,640,638,636,634,629,628,627,626,624,617,615,614,608,606,604,602,600,598,597,596,595,593,586,585,584,583,582,581,580,578,575,570,569,568,566,565,563,552,551,540,534,522,517,516,513,496,495,494,492,491,487,486,485,484,481,480,479,477,476,474,473,467,466,449,425,424,423,414,406,396,395,383,382,367,355,354,336,334,328,324,323,320,317,299,293,289,264,263,258,241,239,233]) ).
thf(642,plain,
$false,
inference(solved_all_splits,[solved_all_splits(join,[])],[641]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET080-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.04 % Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.22 % Computer : n016.cluster.edu
% 0.09/0.22 % Model : x86_64 x86_64
% 0.09/0.22 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.22 % Memory : 8046.5625MB
% 0.09/0.22 % OS : Linux 6.8.0-71-generic
% 0.09/0.22 % CPULimit : 300
% 0.09/0.22 % WCLimit : 300
% 0.09/0.22 % DateTime : Tue Oct 6 14:34:59 UTC 2026
% 0.09/0.23 % CPUTime :
% 0.09/0.23 Running leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 1.74/0.62
% 1.74/0.62 No.of.Axioms: 92
% 1.74/0.62
% 1.74/0.62 Length.of.Defs: 0
% 1.74/0.62
% 1.74/0.62 Contains.Choice.Funs: false
% 1.74/0.62 ................
% 1.74/0.62
% 1.74/0.62 ********************************
% 1.74/0.62 * All subproblems solved! *
% 1.74/0.62 ********************************
% 1.74/0.62 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:1,axioms:92,ps:0,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:false,expand_extuni:false,foatp:e,atp_timeout:299,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:641,loop_count:0,foatp_calls:1,translation:fof_full)
% 1.74/0.62
% 1.74/0.62 %**** Beginning of derivation protocol ****
% 1.74/0.62 % SZS output start CNFRefutation
% See solution above
% 1.74/0.63
% 1.74/0.63 %**** End of derivation protocol ****
% 1.74/0.63 %**** no. of clauses in derivation: 642 ****
% 1.74/0.63 %**** clause counter: 641 ****
% 1.74/0.63
% 1.74/0.63 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:1,axioms:92,ps:0,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:false,expand_extuni:false,foatp:e,atp_timeout:299,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:641,loop_count:0,foatp_calls:1,translation:fof_full)
%------------------------------------------------------------------------------