TPTP Problem File: SET055^12.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SET055^12 : TPTP v8.2.0. Released v8.1.0.
% Domain : Set Theory
% Problem : TPTP problem SET055+1.p with axiomatized equality
% Version : [BP13] axioms.
% English :
% Refs : [RO12] Raths & Otten (2012), The QMLTP Problem Library for Fi
% : [BP13] Benzmueller & Paulson (2013), Quantified Multimodal Lo
% : [Ste22] Steen (2022), An Extensible Logic Embedding Tool for L
% Source : [TPTP]
% Names : SET055+1 [QMLTP]
% Status : Theorem
% Rating : 0.00 v8.2.0, 0.08 v8.1.0
% Syntax : Number of formulae : 142 ( 12 unt; 46 typ; 10 def)
% Number of atoms : 598 ( 10 equ; 0 cnn)
% Maximal formula atoms : 11 ( 6 avg)
% Number of connectives : 1140 ( 1 ~; 1 |; 3 &;1131 @)
% ( 1 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 10 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 106 ( 106 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 45 usr; 6 con; 0-3 aty)
% Number of variables : 226 ( 218 ^; 5 !; 3 ?; 226 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This output was generated by embedproblem, version 1.7.1 (library
% version 1.3). Generated on Thu Apr 28 13:18:18 EDT 2022 using
% 'modal' embedding, version 1.5.2. Logic specification used:
% $modal == [$constants == $rigid,$quantification == $varying,
% $modalities == $modal_system_S5U].
%------------------------------------------------------------------------------
thf(mworld,type,
mworld: $tType ).
thf(mrel_type,type,
mrel: mworld > mworld > $o ).
thf(mactual_type,type,
mactual: mworld ).
thf(mlocal_type,type,
mlocal: ( mworld > $o ) > $o ).
thf(mlocal_def,definition,
( mlocal
= ( ^ [Phi: mworld > $o] : ( Phi @ mactual ) ) ) ).
thf(mnot_type,type,
mnot: ( mworld > $o ) > mworld > $o ).
thf(mand_type,type,
mand: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mor_type,type,
mor: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mimplies_type,type,
mimplies: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mequiv_type,type,
mequiv: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mnot_def,definition,
( mnot
= ( ^ [A: mworld > $o,W: mworld] :
~ ( A @ W ) ) ) ).
thf(mand_def,definition,
( mand
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
& ( B @ W ) ) ) ) ).
thf(mor_def,definition,
( mor
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
| ( B @ W ) ) ) ) ).
thf(mimplies_def,definition,
( mimplies
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
=> ( B @ W ) ) ) ) ).
thf(mequiv_def,definition,
( mequiv
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
<=> ( B @ W ) ) ) ) ).
thf(mbox_type,type,
mbox: ( mworld > $o ) > mworld > $o ).
thf(mbox_def,definition,
( mbox
= ( ^ [Phi: mworld > $o,W: mworld] :
! [V: mworld] :
( ( mrel @ W @ V )
=> ( Phi @ V ) ) ) ) ).
thf(mdia_type,type,
mdia: ( mworld > $o ) > mworld > $o ).
thf(mdia_def,definition,
( mdia
= ( ^ [Phi: mworld > $o,W: mworld] :
? [V: mworld] :
( ( mrel @ W @ V )
& ( Phi @ V ) ) ) ) ).
thf(mrel_universal,axiom,
! [W: mworld,V: mworld] : ( mrel @ W @ V ) ).
thf(eiw_di_type,type,
eiw_di: $i > mworld > $o ).
thf(eiw_di_nonempty,axiom,
! [W: mworld] :
? [X: $i] : ( eiw_di @ X @ W ) ).
thf(mforall_di_type,type,
mforall_di: ( $i > mworld > $o ) > mworld > $o ).
thf(mforall_di_def,definition,
( mforall_di
= ( ^ [A: $i > mworld > $o,W: mworld] :
! [X: $i] :
( ( eiw_di @ X @ W )
=> ( A @ X @ W ) ) ) ) ).
thf(mexists_di_type,type,
mexists_di: ( $i > mworld > $o ) > mworld > $o ).
thf(mexists_di_def,definition,
( mexists_di
= ( ^ [A: $i > mworld > $o,W: mworld] :
? [X: $i] :
( ( eiw_di @ X @ W )
& ( A @ X @ W ) ) ) ) ).
%%% This output was generated by tptputils, version 1.2.
%%% Generated on Wed Apr 27 15:37:57 CEST 2022 using command 'downgrade(tff)'.
thf(universal_class_decl,type,
universal_class: $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(qmltpeq_decl,type,
qmltpeq: $i > $i > mworld > $o ).
thf(inductive_decl,type,
inductive: $i > mworld > $o ).
thf(disjoint_decl,type,
disjoint: $i > $i > mworld > $o ).
thf(function_decl,type,
function: $i > mworld > $o ).
thf(member_decl,type,
member: $i > $i > mworld > $o ).
thf(subclass_decl,type,
subclass: $i > $i > mworld > $o ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(first_decl,type,
first: $i > $i ).
thf(reflexivity,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( qmltpeq @ X @ X ) ) ) ).
thf(symmetry,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( mimplies @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ X ) ) ) ) ) ).
thf(transitivity,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] :
( mforall_di
@ ^ [Z: $i] : ( mimplies @ ( mand @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ Z ) ) @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ).
thf(apply_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( apply @ A @ C ) @ ( apply @ B @ C ) ) ) ) ) ) ) ).
thf(apply_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( apply @ C @ A ) @ ( apply @ C @ B ) ) ) ) ) ) ) ).
thf(complement_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( complement @ A ) @ ( complement @ B ) ) ) ) ) ) ).
thf(compose_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( compose @ A @ C ) @ ( compose @ B @ C ) ) ) ) ) ) ) ).
thf(compose_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( compose @ C @ A ) @ ( compose @ C @ B ) ) ) ) ) ) ) ).
thf(cross_product_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( cross_product @ A @ C ) @ ( cross_product @ B @ C ) ) ) ) ) ) ) ).
thf(cross_product_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( cross_product @ C @ A ) @ ( cross_product @ C @ B ) ) ) ) ) ) ) ).
thf(domain_of_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( domain_of @ A ) @ ( domain_of @ B ) ) ) ) ) ) ).
thf(first_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( first @ A ) @ ( first @ B ) ) ) ) ) ) ).
thf(flip_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( flip @ A ) @ ( flip @ B ) ) ) ) ) ) ).
thf(image_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( image @ A @ C ) @ ( image @ B @ C ) ) ) ) ) ) ) ).
thf(image_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( image @ C @ A ) @ ( image @ C @ B ) ) ) ) ) ) ) ).
thf(intersection_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ A @ C ) @ ( intersection @ B @ C ) ) ) ) ) ) ) ).
thf(intersection_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ C @ A ) @ ( intersection @ C @ B ) ) ) ) ) ) ) ).
thf(inverse_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( inverse @ A ) @ ( inverse @ B ) ) ) ) ) ) ).
thf(ordered_pair_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( ordered_pair @ A @ C ) @ ( ordered_pair @ B @ C ) ) ) ) ) ) ) ).
thf(ordered_pair_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( ordered_pair @ C @ A ) @ ( ordered_pair @ C @ B ) ) ) ) ) ) ) ).
thf(power_class_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( power_class @ A ) @ ( power_class @ B ) ) ) ) ) ) ).
thf(range_of_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( range_of @ A ) @ ( range_of @ B ) ) ) ) ) ) ).
thf(restrict_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] :
( mforall_di
@ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ A @ C @ D ) @ ( restrict @ B @ C @ D ) ) ) ) ) ) ) ) ).
thf(restrict_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] :
( mforall_di
@ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ C @ A @ D ) @ ( restrict @ C @ B @ D ) ) ) ) ) ) ) ) ).
thf(restrict_substitution_3,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] :
( mforall_di
@ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ C @ D @ A ) @ ( restrict @ C @ D @ B ) ) ) ) ) ) ) ) ).
thf(rotate_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( rotate @ A ) @ ( rotate @ B ) ) ) ) ) ) ).
thf(second_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( second @ A ) @ ( second @ B ) ) ) ) ) ) ).
thf(singleton_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) ) ) ) ) ) ).
thf(successor_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( successor @ A ) @ ( successor @ B ) ) ) ) ) ) ).
thf(sum_class_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( sum_class @ A ) @ ( sum_class @ B ) ) ) ) ) ) ).
thf(union_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) ) ) ) ) ) ) ).
thf(union_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) ) ) ) ) ) ) ).
thf(unordered_pair_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ A @ C ) @ ( unordered_pair @ B @ C ) ) ) ) ) ) ) ).
thf(unordered_pair_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ C @ A ) @ ( unordered_pair @ C @ B ) ) ) ) ) ) ) ).
thf(disjoint_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ A @ C ) ) @ ( disjoint @ B @ C ) ) ) ) ) ) ).
thf(disjoint_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ C @ A ) ) @ ( disjoint @ C @ B ) ) ) ) ) ) ).
thf(function_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( function @ A ) ) @ ( function @ B ) ) ) ) ) ).
thf(inductive_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( inductive @ A ) ) @ ( inductive @ B ) ) ) ) ) ).
thf(member_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ A @ C ) ) @ ( member @ B @ C ) ) ) ) ) ) ).
thf(member_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ C @ A ) ) @ ( member @ C @ B ) ) ) ) ) ) ).
thf(subclass_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subclass @ A @ C ) ) @ ( subclass @ B @ C ) ) ) ) ) ) ).
thf(subclass_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subclass @ C @ A ) ) @ ( subclass @ C @ B ) ) ) ) ) ) ).
thf(subclass_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] :
( mequiv @ ( subclass @ X @ Y )
@ ( mforall_di
@ ^ [U: $i] : ( mimplies @ ( member @ U @ X ) @ ( member @ U @ Y ) ) ) ) ) ) ) ).
thf(class_elements_are_sets,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( subclass @ X @ universal_class ) ) ) ).
thf(extensionality,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( mequiv @ ( qmltpeq @ X @ Y ) @ ( mand @ ( subclass @ X @ Y ) @ ( subclass @ Y @ X ) ) ) ) ) ) ).
thf(unordered_pair_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [U: $i] :
( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( mequiv @ ( member @ U @ ( unordered_pair @ X @ Y ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( mor @ ( qmltpeq @ U @ X ) @ ( qmltpeq @ U @ Y ) ) ) ) ) ) ) ) ).
thf(unordered_pair_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ) ) ) ) ).
thf(singleton_set_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( qmltpeq @ ( singleton @ X ) @ ( unordered_pair @ X @ X ) ) ) ) ).
thf(ordered_pair_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( qmltpeq @ ( ordered_pair @ X @ Y ) @ ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) ) ) ) ) ) ).
thf(cross_product_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [U: $i] :
( mforall_di
@ ^ [V: $i] :
( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( mequiv @ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) ) @ ( mand @ ( member @ U @ X ) @ ( member @ V @ Y ) ) ) ) ) ) ) ) ).
thf(cross_product_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] :
( mforall_di
@ ^ [Z: $i] : ( mimplies @ ( member @ Z @ ( cross_product @ X @ Y ) ) @ ( qmltpeq @ Z @ ( ordered_pair @ ( first @ Z ) @ ( second @ Z ) ) ) ) ) ) ) ) ).
thf(element_relation_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( mequiv @ ( member @ ( ordered_pair @ X @ Y ) @ element_relation ) @ ( mand @ ( member @ Y @ universal_class ) @ ( member @ X @ Y ) ) ) ) ) ) ).
thf(element_relation_0,axiom,
mlocal @ ( subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ) ) ).
thf(intersection_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] :
( mforall_di
@ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( intersection @ X @ Y ) ) @ ( mand @ ( member @ Z @ X ) @ ( member @ Z @ Y ) ) ) ) ) ) ) ).
thf(complement_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( complement @ X ) ) @ ( mand @ ( member @ Z @ universal_class ) @ ( mnot @ ( member @ Z @ X ) ) ) ) ) ) ) ).
thf(restrict_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [XR: $i] :
( mforall_di
@ ^ [Y: $i] : ( qmltpeq @ ( restrict @ XR @ X @ Y ) @ ( intersection @ XR @ ( cross_product @ X @ Y ) ) ) ) ) ) ) ).
thf(null_class_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( mnot @ ( member @ X @ null_class ) ) ) ) ).
thf(domain_of_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Z: $i] : ( 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,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [U: $i] :
( mforall_di
@ ^ [V: $i] :
( mforall_di
@ ^ [W: $i] : ( 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_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) ) ) ).
thf(flip_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [U: $i] :
( mforall_di
@ ^ [V: $i] :
( mforall_di
@ ^ [W: $i] :
( mforall_di
@ ^ [X: $i] : ( 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_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) ) ) ).
thf(union_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] :
( mforall_di
@ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( union @ X @ Y ) ) @ ( mor @ ( member @ Z @ X ) @ ( member @ Z @ Y ) ) ) ) ) ) ) ).
thf(successor_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( qmltpeq @ ( successor @ X ) @ ( union @ X @ ( singleton @ X ) ) ) ) ) ).
thf(successor_relation_defn1,axiom,
mlocal @ ( subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ) ) ).
thf(successor_relation_defn2,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( 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,
( mlocal
@ ( mforall_di
@ ^ [Y: $i] : ( qmltpeq @ ( inverse @ Y ) @ ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) ) ) ) ) ).
thf(range_of_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [Z: $i] : ( qmltpeq @ ( range_of @ Z ) @ ( domain_of @ ( inverse @ Z ) ) ) ) ) ).
thf(image_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [XR: $i] : ( qmltpeq @ ( image @ XR @ X ) @ ( range_of @ ( restrict @ XR @ X @ universal_class ) ) ) ) ) ) ).
thf(inductive_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( mequiv @ ( inductive @ X ) @ ( mand @ ( member @ null_class @ X ) @ ( subclass @ ( image @ successor_relation @ X ) @ X ) ) ) ) ) ).
thf(infinity,axiom,
( mlocal
@ ( mexists_di
@ ^ [X: $i] :
( mand @ ( member @ X @ universal_class )
@ ( mand @ ( inductive @ X )
@ ( mforall_di
@ ^ [Y: $i] : ( mimplies @ ( inductive @ Y ) @ ( subclass @ X @ Y ) ) ) ) ) ) ) ).
thf(sum_class_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [U: $i] :
( mforall_di
@ ^ [X: $i] :
( mequiv @ ( member @ U @ ( sum_class @ X ) )
@ ( mexists_di
@ ^ [Y: $i] : ( mand @ ( member @ U @ Y ) @ ( member @ Y @ X ) ) ) ) ) ) ) ).
thf(sum_class_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( mimplies @ ( member @ X @ universal_class ) @ ( member @ ( sum_class @ X ) @ universal_class ) ) ) ) ).
thf(power_class_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [U: $i] :
( mforall_di
@ ^ [X: $i] : ( mequiv @ ( member @ U @ ( power_class @ X ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( subclass @ U @ X ) ) ) ) ) ) ).
thf(power_class_0,axiom,
( mlocal
@ ( mforall_di
@ ^ [U: $i] : ( mimplies @ ( member @ U @ universal_class ) @ ( member @ ( power_class @ U ) @ universal_class ) ) ) ) ).
thf(compose_defn1,axiom,
( mlocal
@ ( mforall_di
@ ^ [XR: $i] :
( mforall_di
@ ^ [YR: $i] : ( subclass @ ( compose @ YR @ XR ) @ ( cross_product @ universal_class @ universal_class ) ) ) ) ) ).
thf(compose_defn2,axiom,
( mlocal
@ ( mforall_di
@ ^ [XR: $i] :
( mforall_di
@ ^ [YR: $i] :
( mforall_di
@ ^ [U: $i] :
( mforall_di
@ ^ [V: $i] : ( 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,
( mlocal
@ ( mforall_di
@ ^ [XF: $i] : ( mequiv @ ( function @ XF ) @ ( mand @ ( subclass @ XF @ ( cross_product @ universal_class @ universal_class ) ) @ ( subclass @ ( compose @ XF @ ( inverse @ XF ) ) @ identity_relation ) ) ) ) ) ).
thf(replacement,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [XF: $i] : ( mimplies @ ( mand @ ( member @ X @ universal_class ) @ ( function @ XF ) ) @ ( member @ ( image @ XF @ X ) @ universal_class ) ) ) ) ) ).
thf(disjoint_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] :
( mequiv @ ( disjoint @ X @ Y )
@ ( mforall_di
@ ^ [U: $i] : ( mnot @ ( mand @ ( member @ U @ X ) @ ( member @ U @ Y ) ) ) ) ) ) ) ) ).
thf(regularity,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mimplies @ ( mnot @ ( qmltpeq @ X @ null_class ) )
@ ( mexists_di
@ ^ [U: $i] : ( mand @ ( member @ U @ universal_class ) @ ( mand @ ( member @ U @ X ) @ ( disjoint @ U @ X ) ) ) ) ) ) ) ).
thf(apply_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [XF: $i] :
( mforall_di
@ ^ [Y: $i] : ( qmltpeq @ ( apply @ XF @ Y ) @ ( sum_class @ ( image @ XF @ ( singleton @ Y ) ) ) ) ) ) ) ).
thf(choice,axiom,
( mlocal
@ ( mexists_di
@ ^ [XF: $i] :
( mand @ ( function @ XF )
@ ( mforall_di
@ ^ [Y: $i] : ( mimplies @ ( member @ Y @ universal_class ) @ ( mor @ ( qmltpeq @ Y @ null_class ) @ ( member @ ( apply @ XF @ Y ) @ Y ) ) ) ) ) ) ) ).
thf(reflexivity_0,conjecture,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( qmltpeq @ X @ X ) ) ) ).
%------------------------------------------------------------------------------