TPTP Problem File: LCL965_15.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL965_15 : TPTP v8.2.0. Released v8.2.0.
% Domain : Set Theory
% Problem : Goedel translation of SET027+1
% Version : [QMLTP] axioms.
% English :
% Refs : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic
% : [RO12] Raths & Otten (2012), The QMLTP Problem Library for Fi
% Source : [QMLTP]
% Names : GSE027+1 [QMLTP]
% Status : CounterSatisfiable
% Rating : 0.33 v8.2.0
% Syntax : Number of formulae : 123 ( 2 unt; 36 typ; 0 def)
% Number of atoms : 983 ( 0 equ)
% Maximal formula atoms : 24 ( 11 avg)
% Number of connectives : 904 ( 8 ~; 5 |; 74 &)
% ( 0 <=>; 817 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 15 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 51 ( 29 >; 22 *; 0 +; 0 <<)
% Number of predicates : 8 ( 8 usr; 0 prp; 2-3 aty)
% Number of functors : 27 ( 27 usr; 6 con; 0-3 aty)
% Number of variables : 741 (; 734 !; 7 ?; 741 :)
% SPC : TF0_CSA_NEQ_NAR
% Comments : Generated by embedproblem, version 1.7.14, rigid constant,
% modal_system_D, TFF embedding.
%------------------------------------------------------------------------------
tff('$ki_world_type',type,
'$ki_world': $tType ).
tff('$ki_local_world_decl',type,
'$ki_local_world': '$ki_world' ).
tff('$ki_accessible_decl',type,
'$ki_accessible': ( '$ki_world' * '$ki_world' ) > $o ).
tff(mrel_serial,axiom,
! [W: '$ki_world'] :
? [V: '$ki_world'] : '$ki_accessible'(W,V) ).
tff(universal_class_decl,type,
universal_class: $i ).
tff(identity_relation_decl,type,
identity_relation: $i ).
tff(successor_relation_decl,type,
successor_relation: $i ).
tff(element_relation_decl,type,
element_relation: $i ).
tff(null_class_decl,type,
null_class: $i ).
tff(qmltpeq_decl,type,
qmltpeq: ( '$ki_world' * $i * $i ) > $o ).
tff(inductive_decl,type,
inductive: ( '$ki_world' * $i ) > $o ).
tff(disjoint_decl,type,
disjoint: ( '$ki_world' * $i * $i ) > $o ).
tff(function_decl,type,
function: ( '$ki_world' * $i ) > $o ).
tff(member_decl,type,
member: ( '$ki_world' * $i * $i ) > $o ).
tff(subclass_decl,type,
subclass: ( '$ki_world' * $i * $i ) > $o ).
tff(singleton_decl,type,
singleton: $i > $i ).
tff(image_decl,type,
image: ( $i * $i ) > $i ).
tff(inverse_decl,type,
inverse: $i > $i ).
tff(rotate_decl,type,
rotate: $i > $i ).
tff(unordered_pair_decl,type,
unordered_pair: ( $i * $i ) > $i ).
tff(successor_decl,type,
successor: $i > $i ).
tff(apply_decl,type,
apply: ( $i * $i ) > $i ).
tff(power_class_decl,type,
power_class: $i > $i ).
tff(range_of_decl,type,
range_of: $i > $i ).
tff(union_decl,type,
union: ( $i * $i ) > $i ).
tff(restrict_decl,type,
restrict: ( $i * $i * $i ) > $i ).
tff(second_decl,type,
second: $i > $i ).
tff(ordered_pair_decl,type,
ordered_pair: ( $i * $i ) > $i ).
tff(domain_of_decl,type,
domain_of: $i > $i ).
tff(sum_class_decl,type,
sum_class: $i > $i ).
tff(compose_decl,type,
compose: ( $i * $i ) > $i ).
tff(intersection_decl,type,
intersection: ( $i * $i ) > $i ).
tff(cross_product_decl,type,
cross_product: ( $i * $i ) > $i ).
tff(complement_decl,type,
complement: $i > $i ).
tff(flip_decl,type,
flip: $i > $i ).
tff(first_decl,type,
first: $i > $i ).
tff('$ki_exists_in_world_$i_decl',type,
'$ki_exists_in_world_$i': ( '$ki_world' * $i ) > $o ).
tff('$ki_exists_in_world_$i_cumul',axiom,
! [W: '$ki_world',V: '$ki_world',X: $i] :
( ( '$ki_exists_in_world_$i'(W,X)
& '$ki_accessible'(W,V) )
=> '$ki_exists_in_world_$i'(V,X) ) ).
tff('$ki_exists_in_world_$i_nonempty',axiom,
! [W: '$ki_world'] :
? [X: $i] : '$ki_exists_in_world_$i'(W,X) ).
tff(reflexivity,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> qmltpeq(W0,X,X) ) ) ) ).
tff(symmetry,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,Y,X) ) ) ) ) ) ) ) ).
tff(transitivity,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W1,Z)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,Y) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,Y,Z) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,Z) ) ) ) ) ) ) ) ) ) ).
tff(apply_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,apply(A,C),apply(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(apply_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,apply(C,A),apply(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(complement_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,complement(A),complement(B)) ) ) ) ) ) ) ) ).
tff(compose_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,compose(A,C),compose(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(compose_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,compose(C,A),compose(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(cross_product_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,cross_product(A,C),cross_product(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(cross_product_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,cross_product(C,A),cross_product(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(domain_of_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,domain_of(A),domain_of(B)) ) ) ) ) ) ) ) ).
tff(first_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,first(A),first(B)) ) ) ) ) ) ) ) ).
tff(flip_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,flip(A),flip(B)) ) ) ) ) ) ) ) ).
tff(image_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,image(A,C),image(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(image_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,image(C,A),image(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(intersection_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,intersection(A,C),intersection(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(intersection_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,intersection(C,A),intersection(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(inverse_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,inverse(A),inverse(B)) ) ) ) ) ) ) ) ).
tff(ordered_pair_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,ordered_pair(A,C),ordered_pair(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(ordered_pair_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,ordered_pair(C,A),ordered_pair(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(power_class_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,power_class(A),power_class(B)) ) ) ) ) ) ) ) ).
tff(range_of_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,range_of(A),range_of(B)) ) ) ) ) ) ) ) ).
tff(restrict_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> qmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> qmltpeq(W4,restrict(A,C,D),restrict(B,C,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(restrict_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> qmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> qmltpeq(W4,restrict(C,A,D),restrict(C,B,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(restrict_substitution_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> qmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> qmltpeq(W4,restrict(C,D,A),restrict(C,D,B)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(rotate_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,rotate(A),rotate(B)) ) ) ) ) ) ) ) ).
tff(second_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,second(A),second(B)) ) ) ) ) ) ) ) ).
tff(singleton_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,singleton(A),singleton(B)) ) ) ) ) ) ) ) ).
tff(successor_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,successor(A),successor(B)) ) ) ) ) ) ) ) ).
tff(sum_class_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,sum_class(A),sum_class(B)) ) ) ) ) ) ) ) ).
tff(union_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,union(A,C),union(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(union_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,union(C,A),union(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(unordered_pair_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,unordered_pair(A,C),unordered_pair(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(unordered_pair_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,unordered_pair(C,A),unordered_pair(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(disjoint_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> disjoint(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> disjoint(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(disjoint_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> disjoint(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> disjoint(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(function_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> function(W2,A) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> function(W2,B) ) ) ) ) ) ) ) ).
tff(inductive_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,A,B) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> inductive(W2,A) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> inductive(W2,B) ) ) ) ) ) ) ) ).
tff(member_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(member_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(subclass_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subclass(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subclass(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(subclass_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subclass(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subclass(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(subclass_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subclass(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W2,U)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,X) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,Y) ) ) ) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W2,U)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,X) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,Y) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subclass(W2,X,Y) ) ) ) ) ) ) ) ) ).
tff(class_elements_are_sets,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> subclass(W0,X,universal_class) ) ) ) ).
tff(extensionality,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,X,Y) )
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subclass(W2,X,Y) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subclass(W2,Y,X) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subclass(W2,X,Y) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subclass(W2,Y,X) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,X,Y) ) ) ) ) ) ) ) ) ).
tff(unordered_pair_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W,U)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,U,unordered_pair(X,Y)) )
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,U,universal_class) )
& ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,U,X) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,U,Y) ) ) ) ) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,U,universal_class) )
& ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,U,X) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,U,Y) ) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,U,unordered_pair(X,Y)) ) ) ) ) ) ) ) ) ) ) ).
tff(unordered_pair_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,unordered_pair(X,Y),universal_class) ) ) ) ) ) ).
tff(singleton_set_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> qmltpeq(W0,singleton(X),unordered_pair(X,X)) ) ) ) ).
tff(ordered_pair_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> qmltpeq(W1,ordered_pair(X,Y),unordered_pair(singleton(X),unordered_pair(X,singleton(Y)))) ) ) ) ) ) ).
tff(cross_product_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W,U)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [V: $i] :
( '$ki_exists_in_world_$i'(W0,V)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W1,X)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W2,Y)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(U,V),cross_product(X,Y)) )
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,X) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,V,Y) ) ) ) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,X) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,V,Y) ) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(U,V),cross_product(X,Y)) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(cross_product_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W1,Z)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,cross_product(X,Y)) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,Z,ordered_pair(first(Z),second(Z))) ) ) ) ) ) ) ) ) ) ).
tff(element_relation_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,ordered_pair(X,Y),element_relation) )
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Y,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,Y) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Y,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,Y) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,ordered_pair(X,Y),element_relation) ) ) ) ) ) ) ) ) ).
tff(element_relation_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> subclass(W,element_relation,cross_product(universal_class,universal_class)) ) ).
tff(intersection_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W1,Z)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,intersection(X,Y)) )
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,X) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,Y) ) ) ) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,X) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,Y) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,intersection(X,Y)) ) ) ) ) ) ) ) ) ) ) ).
tff(complement_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W0,Z)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Z,complement(X)) )
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Z,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ~ ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,X) ) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Z,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ~ ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,X) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Z,complement(X)) ) ) ) ) ) ) ) ) ).
tff(restrict_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [XR: $i] :
( '$ki_exists_in_world_$i'(W0,XR)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,restrict(XR,X,Y),intersection(XR,cross_product(X,Y))) ) ) ) ) ) ) ) ).
tff(null_class_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ~ ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,X,null_class) ) ) ) ) ).
tff(domain_of_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W0,Z)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Z,domain_of(X)) )
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Z,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ~ ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,restrict(X,singleton(Z),universal_class),null_class) ) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Z,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ~ ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,restrict(X,singleton(Z),universal_class),null_class) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Z,domain_of(X)) ) ) ) ) ) ) ) ) ).
tff(rotate_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W0,U)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [V: $i] :
( '$ki_exists_in_world_$i'(W1,V)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [W: $i] :
( '$ki_exists_in_world_$i'(W2,W)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(U,V),W),rotate(X)) )
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(U,V),W),cross_product(cross_product(universal_class,universal_class),universal_class)) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(V,W),U),X) ) ) ) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(U,V),W),cross_product(cross_product(universal_class,universal_class),universal_class)) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(V,W),U),X) ) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(U,V),W),rotate(X)) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(rotate_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> subclass(W0,rotate(X),cross_product(cross_product(universal_class,universal_class),universal_class)) ) ) ) ).
tff(flip_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W,U)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [V: $i] :
( '$ki_exists_in_world_$i'(W0,V)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [W: $i] :
( '$ki_exists_in_world_$i'(W1,W)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W2,X)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(U,V),W),flip(X)) )
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(U,V),W),cross_product(cross_product(universal_class,universal_class),universal_class)) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(V,U),W),X) ) ) ) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(U,V),W),cross_product(cross_product(universal_class,universal_class),universal_class)) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(V,U),W),X) ) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(ordered_pair(U,V),W),flip(X)) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(flip_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> subclass(W0,flip(X),cross_product(cross_product(universal_class,universal_class),universal_class)) ) ) ) ).
tff(union_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W1,Z)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,union(X,Y)) )
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,X) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,Y) ) ) ) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,X) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,Y) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,Z,union(X,Y)) ) ) ) ) ) ) ) ) ) ) ).
tff(successor_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> qmltpeq(W0,successor(X),union(X,singleton(X))) ) ) ) ).
tff(successor_relation_defn1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> subclass(W,successor_relation,cross_product(universal_class,universal_class)) ) ).
tff(successor_relation_defn2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,ordered_pair(X,Y),successor_relation) )
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Y,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,successor(X),Y) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Y,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,successor(X),Y) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,ordered_pair(X,Y),successor_relation) ) ) ) ) ) ) ) ) ).
tff(inverse_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W,Y)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> qmltpeq(W0,inverse(Y),domain_of(flip(cross_product(Y,universal_class)))) ) ) ) ).
tff(range_of_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W,Z)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> qmltpeq(W0,range_of(Z),domain_of(inverse(Z))) ) ) ) ).
tff(image_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [XR: $i] :
( '$ki_exists_in_world_$i'(W0,XR)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> qmltpeq(W1,image(XR,X),range_of(restrict(XR,X,universal_class))) ) ) ) ) ) ).
tff(inductive_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> inductive(W1,X) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,null_class,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> subclass(W1,image(successor_relation,X),X) ) ) ) )
& ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,null_class,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> subclass(W1,image(successor_relation,X),X) ) )
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> inductive(W1,X) ) ) ) ) ) ) ).
tff(infinity,axiom,
? [X: $i] :
( '$ki_exists_in_world_$i'('$ki_local_world',X)
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> member(W,X,universal_class) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> inductive(W,X) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W,Y)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> inductive(W1,Y) )
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> subclass(W1,X,Y) ) ) ) ) ) ) ).
tff(sum_class_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W,U)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,U,sum_class(X)) )
=> ? [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,U,Y) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Y,X) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ? [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,U,Y) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Y,X) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,U,sum_class(X)) ) ) ) ) ) ) ) ) ).
tff(sum_class_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,X,universal_class) )
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,sum_class(X),universal_class) ) ) ) ) ) ).
tff(power_class_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W,U)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,U,power_class(X)) )
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,U,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subclass(W2,U,X) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,U,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subclass(W2,U,X) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,U,power_class(X)) ) ) ) ) ) ) ) ) ).
tff(power_class_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W,U)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,U,universal_class) )
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,power_class(U),universal_class) ) ) ) ) ) ).
tff(compose_defn1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [XR: $i] :
( '$ki_exists_in_world_$i'(W,XR)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [YR: $i] :
( '$ki_exists_in_world_$i'(W0,YR)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> subclass(W1,compose(YR,XR),cross_product(universal_class,universal_class)) ) ) ) ) ) ).
tff(compose_defn2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [XR: $i] :
( '$ki_exists_in_world_$i'(W,XR)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [YR: $i] :
( '$ki_exists_in_world_$i'(W0,YR)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W1,U)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [V: $i] :
( '$ki_exists_in_world_$i'(W2,V)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(U,V),compose(YR,XR)) )
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,universal_class) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,V,image(YR,image(YR,singleton(U)))) ) ) ) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,universal_class) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,V,image(YR,image(YR,singleton(U)))) ) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,ordered_pair(U,V),compose(YR,XR)) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(function_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [XF: $i] :
( '$ki_exists_in_world_$i'(W,XF)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> function(W1,XF) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> subclass(W1,XF,cross_product(universal_class,universal_class)) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> subclass(W1,compose(XF,inverse(XF)),identity_relation) ) ) ) )
& ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> subclass(W1,XF,cross_product(universal_class,universal_class)) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> subclass(W1,compose(XF,inverse(XF)),identity_relation) ) )
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> function(W1,XF) ) ) ) ) ) ) ).
tff(replacement,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [XF: $i] :
( '$ki_exists_in_world_$i'(W0,XF)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,universal_class) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> function(W2,XF) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,image(XF,X),universal_class) ) ) ) ) ) ) ) ).
tff(disjoint_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> disjoint(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W2,U)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ~ ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,X) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,Y) ) ) ) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W2,U)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ~ ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,X) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,U,Y) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> disjoint(W2,X,Y) ) ) ) ) ) ) ) ) ).
tff(regularity,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ~ ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,X,null_class) ) )
=> ? [U: $i] :
( '$ki_exists_in_world_$i'(W0,U)
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,U,universal_class) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,U,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> disjoint(W1,U,X) ) ) ) ) ) ) ).
tff(apply_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [XF: $i] :
( '$ki_exists_in_world_$i'(W,XF)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> qmltpeq(W1,apply(XF,Y),sum_class(image(XF,singleton(Y)))) ) ) ) ) ) ).
tff(choice,axiom,
? [XF: $i] :
( '$ki_exists_in_world_$i'('$ki_local_world',XF)
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> function(W,XF) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W,Y)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,Y,universal_class) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> qmltpeq(W1,Y,null_class) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,apply(XF,Y),Y) ) ) ) ) ) ) ) ).
tff(verify,conjecture,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W1,Z)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subclass(W3,X,Y) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subclass(W3,Y,Z) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subclass(W3,X,Z) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------