TPTP Problem File: SET075-7.p
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%--------------------------------------------------------------------------
% File : SET075-7 : TPTP v8.2.0. Bugfixed v2.1.0.
% Domain : Set Theory
% Problem : Corollary to unordered pair axiom
% Version : [Qua92] axioms : Augmented.
% English :
% Refs : [Qua92] Quaife (1992), Automated Deduction in von Neumann-Bern
% Source : [Quaife]
% Names : UP6 cor. [Quaife]
% Status : Unsatisfiable
% Rating : 0.05 v8.2.0, 0.10 v8.1.0, 0.11 v7.4.0, 0.12 v7.3.0, 0.00 v7.0.0, 0.13 v6.4.0, 0.07 v6.3.0, 0.00 v6.2.0, 0.10 v6.1.0, 0.07 v6.0.0, 0.00 v5.5.0, 0.20 v5.3.0, 0.17 v5.2.0, 0.12 v5.1.0, 0.18 v5.0.0, 0.14 v4.1.0, 0.15 v4.0.1, 0.27 v3.7.0, 0.20 v3.5.0, 0.18 v3.4.0, 0.08 v3.3.0, 0.07 v3.2.0, 0.08 v3.1.0, 0.09 v2.7.0, 0.08 v2.6.0, 0.00 v2.1.0
% Syntax : Number of clauses : 117 ( 39 unt; 15 nHn; 78 RR)
% Number of literals : 231 ( 54 equ; 101 neg)
% Maximal clause size : 5 ( 1 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 42 ( 42 usr; 12 con; 0-3 aty)
% Number of variables : 228 ( 40 sgn)
% SPC : CNF_UNS_RFO_SEQ_NHN
% Comments : Preceding lemmas are added.
% : Not in [Qua92].
% Bugfixes : v2.1.0 - Bugfix in SET004-0.ax.
%--------------------------------------------------------------------------
%----Include von Neuman-Bernays-Godel set theory axioms
include('Axioms/SET004-0.ax').
%--------------------------------------------------------------------------
%----Corollaries to Unordered pair axiom. Not in paper, but in email.
cnf(corollary_1_to_unordered_pair,axiom,
( ~ member(ordered_pair(X,Y),cross_product(U,V))
| member(X,unordered_pair(X,Y)) ) ).
cnf(corollary_2_to_unordered_pair,axiom,
( ~ member(ordered_pair(X,Y),cross_product(U,V))
| member(Y,unordered_pair(X,Y)) ) ).
%----Corollaries to Cartesian product axiom.
cnf(corollary_1_to_cartesian_product,axiom,
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,universal_class) ) ).
cnf(corollary_2_to_cartesian_product,axiom,
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ) ).
%---- PARTIAL ORDER.
%----(PO1): reflexive.
cnf(subclass_is_reflexive,axiom,
subclass(X,X) ).
%----(PO2): antisymmetry is part of A-3.
%----(x < y), (y < x) --> (x = y).
%----(PO3): transitivity.
cnf(transitivity_of_subclass,axiom,
( ~ subclass(X,Y)
| ~ subclass(Y,Z)
| subclass(X,Z) ) ).
%---- EQUALITY.
%----(EQ1): equality axiom.
%----a:x:(x = x).
%----This is always an axiom in the TPTP presentation.
%----(EQ2): expanded equality definition.
cnf(equality1,axiom,
( X = Y
| member(not_subclass_element(X,Y),X)
| member(not_subclass_element(Y,X),Y) ) ).
cnf(equality2,axiom,
( ~ member(not_subclass_element(X,Y),Y)
| X = Y
| member(not_subclass_element(Y,X),Y) ) ).
cnf(equality3,axiom,
( ~ member(not_subclass_element(Y,X),X)
| X = Y
| member(not_subclass_element(X,Y),X) ) ).
cnf(equality4,axiom,
( ~ member(not_subclass_element(X,Y),Y)
| ~ member(not_subclass_element(Y,X),X)
| X = Y ) ).
%---- SPECIAL CLASSES.
%----(SP1): lemma.
cnf(special_classes_lemma,axiom,
~ member(Y,intersection(complement(X),X)) ).
%----(SP2): Existence of O (null class).
%----e:x:a:z:(-(z e x)).
cnf(existence_of_null_class,axiom,
~ member(Z,null_class) ).
%----(SP3): O is a subclass of every class.
cnf(null_class_is_subclass,axiom,
subclass(null_class,X) ).
%----corollary.
cnf(corollary_of_null_class_is_subclass,axiom,
( ~ subclass(X,null_class)
| X = null_class ) ).
%----(SP4): uniqueness of null class.
cnf(null_class_is_unique,axiom,
( Z = null_class
| member(not_subclass_element(Z,null_class),Z) ) ).
%----(SP5): O is a set (follows from axiom of infinity).
cnf(null_class_is_a_set,axiom,
member(null_class,universal_class) ).
%---- UNORDERED PAIRS.
%----(UP1): unordered pair is commutative.
cnf(commutativity_of_unordered_pair,axiom,
unordered_pair(X,Y) = unordered_pair(Y,X) ).
%----(UP2): if one argument is a proper class, pair contains only the
%----other. In a slightly different form to the paper
cnf(singleton_in_unordered_pair1,axiom,
subclass(singleton(X),unordered_pair(X,Y)) ).
cnf(singleton_in_unordered_pair2,axiom,
subclass(singleton(Y),unordered_pair(X,Y)) ).
cnf(unordered_pair_equals_singleton1,axiom,
( member(Y,universal_class)
| unordered_pair(X,Y) = singleton(X) ) ).
cnf(unordered_pair_equals_singleton2,axiom,
( member(X,universal_class)
| unordered_pair(X,Y) = singleton(Y) ) ).
%----(UP3): if both arguments are proper classes, pair is null.
cnf(null_unordered_pair,axiom,
( unordered_pair(X,Y) = null_class
| member(X,universal_class)
| member(Y,universal_class) ) ).
%----(UP4): left cancellation for unordered pairs.
cnf(left_cancellation,axiom,
( unordered_pair(X,Y) != unordered_pair(X,Z)
| ~ member(ordered_pair(Y,Z),cross_product(universal_class,universal_class))
| Y = Z ) ).
%----(UP5): right cancellation for unordered pairs.
cnf(right_cancellation,axiom,
( unordered_pair(X,Z) != unordered_pair(Y,Z)
| ~ member(ordered_pair(X,Y),cross_product(universal_class,universal_class))
| X = Y ) ).
cnf(prove_corollary_to_unordered_pair_axiom3_1,negated_conjecture,
member(ordered_pair(x,y),cross_product(u,v)) ).
cnf(prove_corollary_to_unordered_pair_axiom3_2,negated_conjecture,
unordered_pair(x,y) = null_class ).
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