TPTP Problem File: SET018^7.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SET018^7 : TPTP v8.2.0. Released v5.5.0.
% Domain : Set Theory
% Problem : Second components of equal ordered pairs are equal
% Version : [Ben12] axioms.
% English :
% Refs : [Qua92] Quaife (1992), Automated Deduction in von Neumann-Bern
% : [BL+86] Boyer et al. (1986), Set Theory in First-Order Logic:
% : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source : [Ben12]
% Names : s4-cumul-SET018+1 [Ben12]
% Status : Theorem
% Rating : 1.00 v5.5.0
% Syntax : Number of formulae : 213 ( 59 unt; 67 typ; 32 def)
% Number of atoms : 711 ( 36 equ; 0 cnn)
% Maximal formula atoms : 11 ( 4 avg)
% Number of connectives : 1347 ( 5 ~; 5 |; 9 &;1318 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 224 ( 224 >; 0 *; 0 +; 0 <<)
% Number of symbols : 78 ( 76 usr; 16 con; 0-3 aty)
% Number of variables : 345 ( 247 ^; 91 !; 7 ?; 345 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
%------------------------------------------------------------------------------
%----Include axioms for Modal logic S4 under cumulative domains
include('Axioms/LCL015^0.ax').
include('Axioms/LCL013^5.ax').
include('Axioms/LCL015^1.ax').
%------------------------------------------------------------------------------
thf(inductive_type,type,
inductive: mu > $i > $o ).
thf(subclass_type,type,
subclass: mu > mu > $i > $o ).
thf(disjoint_type,type,
disjoint: mu > mu > $i > $o ).
thf(function_type,type,
function: mu > $i > $o ).
thf(member_type,type,
member: mu > mu > $i > $o ).
thf(unordered_pair_type,type,
unordered_pair: mu > mu > mu ).
thf(existence_of_unordered_pair_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( unordered_pair @ V2 @ V1 ) @ V ) ).
thf(second_type,type,
second: mu > mu ).
thf(existence_of_second_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( second @ V1 ) @ V ) ).
thf(first_type,type,
first: mu > mu ).
thf(existence_of_first_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( first @ V1 ) @ V ) ).
thf(element_relation_type,type,
element_relation: mu ).
thf(existence_of_element_relation_ax,axiom,
! [V: $i] : ( exists_in_world @ element_relation @ V ) ).
thf(complement_type,type,
complement: mu > mu ).
thf(existence_of_complement_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( complement @ V1 ) @ V ) ).
thf(intersection_type,type,
intersection: mu > mu > mu ).
thf(existence_of_intersection_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( intersection @ V2 @ V1 ) @ V ) ).
thf(rotate_type,type,
rotate: mu > mu ).
thf(existence_of_rotate_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( rotate @ V1 ) @ V ) ).
thf(union_type,type,
union: mu > mu > mu ).
thf(existence_of_union_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( union @ V2 @ V1 ) @ V ) ).
thf(successor_type,type,
successor: mu > mu ).
thf(existence_of_successor_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( successor @ V1 ) @ V ) ).
thf(flip_type,type,
flip: mu > mu ).
thf(existence_of_flip_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( flip @ V1 ) @ V ) ).
thf(domain_of_type,type,
domain_of: mu > mu ).
thf(existence_of_domain_of_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( domain_of @ V1 ) @ V ) ).
thf(restrict_type,type,
restrict: mu > mu > mu > mu ).
thf(existence_of_restrict_ax,axiom,
! [V: $i,V3: mu,V2: mu,V1: mu] : ( exists_in_world @ ( restrict @ V3 @ V2 @ V1 ) @ V ) ).
thf(range_of_type,type,
range_of: mu > mu ).
thf(existence_of_range_of_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( range_of @ V1 ) @ V ) ).
thf(successor_relation_type,type,
successor_relation: mu ).
thf(existence_of_successor_relation_ax,axiom,
! [V: $i] : ( exists_in_world @ successor_relation @ V ) ).
thf(power_class_type,type,
power_class: mu > mu ).
thf(existence_of_power_class_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( power_class @ V1 ) @ V ) ).
thf(identity_relation_type,type,
identity_relation: mu ).
thf(existence_of_identity_relation_ax,axiom,
! [V: $i] : ( exists_in_world @ identity_relation @ V ) ).
thf(inverse_type,type,
inverse: mu > mu ).
thf(existence_of_inverse_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( inverse @ V1 ) @ V ) ).
thf(compose_type,type,
compose: mu > mu > mu ).
thf(existence_of_compose_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( compose @ V2 @ V1 ) @ V ) ).
thf(cross_product_type,type,
cross_product: mu > mu > mu ).
thf(existence_of_cross_product_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( cross_product @ V2 @ V1 ) @ V ) ).
thf(singleton_type,type,
singleton: mu > mu ).
thf(existence_of_singleton_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( singleton @ V1 ) @ V ) ).
thf(image_type,type,
image: mu > mu > mu ).
thf(existence_of_image_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( image @ V2 @ V1 ) @ V ) ).
thf(sum_class_type,type,
sum_class: mu > mu ).
thf(existence_of_sum_class_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( sum_class @ V1 ) @ V ) ).
thf(apply_type,type,
apply: mu > mu > mu ).
thf(existence_of_apply_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( apply @ V2 @ V1 ) @ V ) ).
thf(null_class_type,type,
null_class: mu ).
thf(existence_of_null_class_ax,axiom,
! [V: $i] : ( exists_in_world @ null_class @ V ) ).
thf(universal_class_type,type,
universal_class: mu ).
thf(existence_of_universal_class_ax,axiom,
! [V: $i] : ( exists_in_world @ universal_class @ V ) ).
thf(ordered_pair_type,type,
ordered_pair: mu > mu > mu ).
thf(existence_of_ordered_pair_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( ordered_pair @ V2 @ V1 ) @ V ) ).
thf(reflexivity,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( qmltpeq @ X @ X ) ) ) ).
thf(symmetry,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( mimplies @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ X ) ) ) ) ) ).
thf(transitivity,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] :
( mforall_ind
@ ^ [Z: mu] : ( mimplies @ ( mand @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ Z ) ) @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ).
thf(apply_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( apply @ A @ C ) @ ( apply @ B @ C ) ) ) ) ) ) ) ).
thf(apply_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( apply @ C @ A ) @ ( apply @ C @ B ) ) ) ) ) ) ) ).
thf(complement_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( complement @ A ) @ ( complement @ B ) ) ) ) ) ) ).
thf(compose_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( compose @ A @ C ) @ ( compose @ B @ C ) ) ) ) ) ) ) ).
thf(compose_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( compose @ C @ A ) @ ( compose @ C @ B ) ) ) ) ) ) ) ).
thf(cross_product_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( cross_product @ A @ C ) @ ( cross_product @ B @ C ) ) ) ) ) ) ) ).
thf(cross_product_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( cross_product @ C @ A ) @ ( cross_product @ C @ B ) ) ) ) ) ) ) ).
thf(domain_of_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( domain_of @ A ) @ ( domain_of @ B ) ) ) ) ) ) ).
thf(first_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( first @ A ) @ ( first @ B ) ) ) ) ) ) ).
thf(flip_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( flip @ A ) @ ( flip @ B ) ) ) ) ) ) ).
thf(image_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( image @ A @ C ) @ ( image @ B @ C ) ) ) ) ) ) ) ).
thf(image_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( image @ C @ A ) @ ( image @ C @ B ) ) ) ) ) ) ) ).
thf(intersection_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ A @ C ) @ ( intersection @ B @ C ) ) ) ) ) ) ) ).
thf(intersection_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ C @ A ) @ ( intersection @ C @ B ) ) ) ) ) ) ) ).
thf(inverse_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( inverse @ A ) @ ( inverse @ B ) ) ) ) ) ) ).
thf(ordered_pair_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( ordered_pair @ A @ C ) @ ( ordered_pair @ B @ C ) ) ) ) ) ) ) ).
thf(ordered_pair_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( ordered_pair @ C @ A ) @ ( ordered_pair @ C @ B ) ) ) ) ) ) ) ).
thf(power_class_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( power_class @ A ) @ ( power_class @ B ) ) ) ) ) ) ).
thf(range_of_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( range_of @ A ) @ ( range_of @ B ) ) ) ) ) ) ).
thf(restrict_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] :
( mforall_ind
@ ^ [D: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ A @ C @ D ) @ ( restrict @ B @ C @ D ) ) ) ) ) ) ) ) ).
thf(restrict_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] :
( mforall_ind
@ ^ [D: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ C @ A @ D ) @ ( restrict @ C @ B @ D ) ) ) ) ) ) ) ) ).
thf(restrict_substitution_3,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] :
( mforall_ind
@ ^ [D: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ C @ D @ A ) @ ( restrict @ C @ D @ B ) ) ) ) ) ) ) ) ).
thf(rotate_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( rotate @ A ) @ ( rotate @ B ) ) ) ) ) ) ).
thf(second_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( second @ A ) @ ( second @ B ) ) ) ) ) ) ).
thf(singleton_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) ) ) ) ) ) ).
thf(successor_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( successor @ A ) @ ( successor @ B ) ) ) ) ) ) ).
thf(sum_class_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( sum_class @ A ) @ ( sum_class @ B ) ) ) ) ) ) ).
thf(union_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) ) ) ) ) ) ) ).
thf(union_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) ) ) ) ) ) ) ).
thf(unordered_pair_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ A @ C ) @ ( unordered_pair @ B @ C ) ) ) ) ) ) ) ).
thf(unordered_pair_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ C @ A ) @ ( unordered_pair @ C @ B ) ) ) ) ) ) ) ).
thf(disjoint_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ A @ C ) ) @ ( disjoint @ B @ C ) ) ) ) ) ) ).
thf(disjoint_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ C @ A ) ) @ ( disjoint @ C @ B ) ) ) ) ) ) ).
thf(function_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( function @ A ) ) @ ( function @ B ) ) ) ) ) ).
thf(inductive_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( inductive @ A ) ) @ ( inductive @ B ) ) ) ) ) ).
thf(member_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ A @ C ) ) @ ( member @ B @ C ) ) ) ) ) ) ).
thf(member_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ C @ A ) ) @ ( member @ C @ B ) ) ) ) ) ) ).
thf(subclass_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subclass @ A @ C ) ) @ ( subclass @ B @ C ) ) ) ) ) ) ).
thf(subclass_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subclass @ C @ A ) ) @ ( subclass @ C @ B ) ) ) ) ) ) ).
thf(subclass_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] :
( mequiv @ ( subclass @ X @ Y )
@ ( mforall_ind
@ ^ [U: mu] : ( mimplies @ ( member @ U @ X ) @ ( member @ U @ Y ) ) ) ) ) ) ) ).
thf(class_elements_are_sets,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( subclass @ X @ universal_class ) ) ) ).
thf(extensionality,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( mequiv @ ( qmltpeq @ X @ Y ) @ ( mand @ ( subclass @ X @ Y ) @ ( subclass @ Y @ X ) ) ) ) ) ) ).
thf(unordered_pair_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [U: mu] :
( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( mequiv @ ( member @ U @ ( unordered_pair @ X @ Y ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( mor @ ( qmltpeq @ U @ X ) @ ( qmltpeq @ U @ Y ) ) ) ) ) ) ) ) ).
thf(unordered_pair,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ) ) ) ) ).
thf(singleton_set_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( qmltpeq @ ( singleton @ X ) @ ( unordered_pair @ X @ X ) ) ) ) ).
thf(ordered_pair_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( qmltpeq @ ( ordered_pair @ X @ Y ) @ ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) ) ) ) ) ) ).
thf(cross_product_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [U: mu] :
( mforall_ind
@ ^ [V: mu] :
( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( mequiv @ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) ) @ ( mand @ ( member @ U @ X ) @ ( member @ V @ Y ) ) ) ) ) ) ) ) ).
thf(cross_product,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] :
( mforall_ind
@ ^ [Z: mu] : ( mimplies @ ( member @ Z @ ( cross_product @ X @ Y ) ) @ ( qmltpeq @ Z @ ( ordered_pair @ ( first @ Z ) @ ( second @ Z ) ) ) ) ) ) ) ) ).
thf(element_relation_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( mequiv @ ( member @ ( ordered_pair @ X @ Y ) @ element_relation ) @ ( mand @ ( member @ Y @ universal_class ) @ ( member @ X @ Y ) ) ) ) ) ) ).
thf(element_relation,axiom,
mvalid @ ( subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ) ) ).
thf(intersection,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] :
( mforall_ind
@ ^ [Z: mu] : ( mequiv @ ( member @ Z @ ( intersection @ X @ Y ) ) @ ( mand @ ( member @ Z @ X ) @ ( member @ Z @ Y ) ) ) ) ) ) ) ).
thf(complement,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Z: mu] : ( mequiv @ ( member @ Z @ ( complement @ X ) ) @ ( mand @ ( member @ Z @ universal_class ) @ ( mnot @ ( member @ Z @ X ) ) ) ) ) ) ) ).
thf(restrict_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [XR: mu] :
( mforall_ind
@ ^ [Y: mu] : ( qmltpeq @ ( restrict @ XR @ X @ Y ) @ ( intersection @ XR @ ( cross_product @ X @ Y ) ) ) ) ) ) ) ).
thf(null_class_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( mnot @ ( member @ X @ null_class ) ) ) ) ).
thf(domain_of,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Z: mu] : ( mequiv @ ( member @ Z @ ( domain_of @ X ) ) @ ( mand @ ( member @ Z @ universal_class ) @ ( mnot @ ( qmltpeq @ ( restrict @ X @ ( singleton @ Z ) @ universal_class ) @ null_class ) ) ) ) ) ) ) ).
thf(rotate_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [U: mu] :
( mforall_ind
@ ^ [V: mu] :
( mforall_ind
@ ^ [W: mu] : ( mequiv @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) ) @ ( mand @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) @ ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X ) ) ) ) ) ) ) ) ).
thf(rotate,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) ) ) ).
thf(flip_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [U: mu] :
( mforall_ind
@ ^ [V: mu] :
( mforall_ind
@ ^ [W: mu] :
( mforall_ind
@ ^ [X: mu] : ( mequiv @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) ) @ ( mand @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) @ ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X ) ) ) ) ) ) ) ) ).
thf(flip,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) ) ) ).
thf(union_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] :
( mforall_ind
@ ^ [Z: mu] : ( mequiv @ ( member @ Z @ ( union @ X @ Y ) ) @ ( mor @ ( member @ Z @ X ) @ ( member @ Z @ Y ) ) ) ) ) ) ) ).
thf(successor_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( qmltpeq @ ( successor @ X ) @ ( union @ X @ ( singleton @ X ) ) ) ) ) ).
thf(successor_relation_defn1,axiom,
mvalid @ ( subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ) ) ).
thf(successor_relation_defn2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( mequiv @ ( member @ ( ordered_pair @ X @ Y ) @ successor_relation ) @ ( mand @ ( member @ X @ universal_class ) @ ( mand @ ( member @ Y @ universal_class ) @ ( qmltpeq @ ( successor @ X ) @ Y ) ) ) ) ) ) ) ).
thf(inverse_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [Y: mu] : ( qmltpeq @ ( inverse @ Y ) @ ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) ) ) ) ) ).
thf(range_of_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [Z: mu] : ( qmltpeq @ ( range_of @ Z ) @ ( domain_of @ ( inverse @ Z ) ) ) ) ) ).
thf(image_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [XR: mu] : ( qmltpeq @ ( image @ XR @ X ) @ ( range_of @ ( restrict @ XR @ X @ universal_class ) ) ) ) ) ) ).
thf(inductive_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( mequiv @ ( inductive @ X ) @ ( mand @ ( member @ null_class @ X ) @ ( subclass @ ( image @ successor_relation @ X ) @ X ) ) ) ) ) ).
thf(infinity,axiom,
( mvalid
@ ( mexists_ind
@ ^ [X: mu] :
( mand @ ( member @ X @ universal_class )
@ ( mand @ ( inductive @ X )
@ ( mforall_ind
@ ^ [Y: mu] : ( mimplies @ ( inductive @ Y ) @ ( subclass @ X @ Y ) ) ) ) ) ) ) ).
thf(sum_class_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [U: mu] :
( mforall_ind
@ ^ [X: mu] :
( mequiv @ ( member @ U @ ( sum_class @ X ) )
@ ( mexists_ind
@ ^ [Y: mu] : ( mand @ ( member @ U @ Y ) @ ( member @ Y @ X ) ) ) ) ) ) ) ).
thf(sum_class,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( mimplies @ ( member @ X @ universal_class ) @ ( member @ ( sum_class @ X ) @ universal_class ) ) ) ) ).
thf(power_class_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [U: mu] :
( mforall_ind
@ ^ [X: mu] : ( mequiv @ ( member @ U @ ( power_class @ X ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( subclass @ U @ X ) ) ) ) ) ) ).
thf(power_class,axiom,
( mvalid
@ ( mforall_ind
@ ^ [U: mu] : ( mimplies @ ( member @ U @ universal_class ) @ ( member @ ( power_class @ U ) @ universal_class ) ) ) ) ).
thf(compose_defn1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [XR: mu] :
( mforall_ind
@ ^ [YR: mu] : ( subclass @ ( compose @ YR @ XR ) @ ( cross_product @ universal_class @ universal_class ) ) ) ) ) ).
thf(compose_defn2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [XR: mu] :
( mforall_ind
@ ^ [YR: mu] :
( mforall_ind
@ ^ [U: mu] :
( mforall_ind
@ ^ [V: mu] : ( mequiv @ ( member @ ( ordered_pair @ U @ V ) @ ( compose @ YR @ XR ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( member @ V @ ( image @ YR @ ( image @ YR @ ( singleton @ U ) ) ) ) ) ) ) ) ) ) ) ).
thf(function_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [XF: mu] : ( mequiv @ ( function @ XF ) @ ( mand @ ( subclass @ XF @ ( cross_product @ universal_class @ universal_class ) ) @ ( subclass @ ( compose @ XF @ ( inverse @ XF ) ) @ identity_relation ) ) ) ) ) ).
thf(replacement,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [XF: mu] : ( mimplies @ ( mand @ ( member @ X @ universal_class ) @ ( function @ XF ) ) @ ( member @ ( image @ XF @ X ) @ universal_class ) ) ) ) ) ).
thf(disjoint_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] :
( mequiv @ ( disjoint @ X @ Y )
@ ( mforall_ind
@ ^ [U: mu] : ( mnot @ ( mand @ ( member @ U @ X ) @ ( member @ U @ Y ) ) ) ) ) ) ) ) ).
thf(regularity,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mimplies @ ( mnot @ ( qmltpeq @ X @ null_class ) )
@ ( mexists_ind
@ ^ [U: mu] : ( mand @ ( member @ U @ universal_class ) @ ( mand @ ( member @ U @ X ) @ ( disjoint @ U @ X ) ) ) ) ) ) ) ).
thf(apply_defn,axiom,
( mvalid
@ ( mforall_ind
@ ^ [XF: mu] :
( mforall_ind
@ ^ [Y: mu] : ( qmltpeq @ ( apply @ XF @ Y ) @ ( sum_class @ ( image @ XF @ ( singleton @ Y ) ) ) ) ) ) ) ).
thf(choice,axiom,
( mvalid
@ ( mexists_ind
@ ^ [XF: mu] :
( mand @ ( function @ XF )
@ ( mforall_ind
@ ^ [Y: mu] : ( mimplies @ ( member @ Y @ universal_class ) @ ( mor @ ( qmltpeq @ Y @ null_class ) @ ( member @ ( apply @ XF @ Y ) @ Y ) ) ) ) ) ) ) ).
thf(ordered_pair_determines_components2,conjecture,
( mvalid
@ ( mforall_ind
@ ^ [W: mu] :
( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] :
( mforall_ind
@ ^ [Z: mu] : ( mimplies @ ( mand @ ( qmltpeq @ ( ordered_pair @ W @ X ) @ ( ordered_pair @ Y @ Z ) ) @ ( member @ X @ universal_class ) ) @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------