TPTP Problem File: SET092-7.p
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%--------------------------------------------------------------------------
% File : SET092-7 : TPTP v8.2.0. Bugfixed v2.1.0.
% Domain : Set Theory
% Problem : Member_of(X) is unique if X is not a singleton, part 2
% Version : [Qua92] axioms : Augmented.
% English :
% Refs : [Qua92] Quaife (1992), Automated Deduction in von Neumann-Bern
% Source : [Quaife]
% Names : SS8.2 [Qua92]
% Status : Unknown
% Rating : 1.00 v2.1.0
% Syntax : Number of clauses : 136 ( 42 unt; 19 nHn; 92 RR)
% Number of literals : 271 ( 73 equ; 124 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 : 41 ( 41 usr; 9 con; 0-3 aty)
% Number of variables : 259 ( 46 sgn)
% SPC : CNF_UNK_RFO_SEQ_NHN
% Comments : Preceding lemmas are added.
% 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 ) ).
%----(UP6): corollary to (A-4).
cnf(corollary_to_unordered_pair_axiom1,axiom,
( ~ member(X,universal_class)
| unordered_pair(X,Y) != null_class ) ).
cnf(corollary_to_unordered_pair_axiom2,axiom,
( ~ member(Y,universal_class)
| unordered_pair(X,Y) != null_class ) ).
%----corollary to instantiate variables.
%----Not in the paper
cnf(corollary_to_unordered_pair_axiom3,axiom,
( ~ member(ordered_pair(X,Y),cross_product(U,V))
| unordered_pair(X,Y) != null_class ) ).
%----(UP7): if both members of a pair belong to a set, the pair
%----is a subset.
cnf(unordered_pair_is_subset,axiom,
( ~ member(X,Z)
| ~ member(Y,Z)
| subclass(unordered_pair(X,Y),Z) ) ).
%---- SINGLETONS.
%----(SS1): every singleton is a set.
cnf(singletons_are_sets,axiom,
member(singleton(X),universal_class) ).
%----corollary, not in the paper.
cnf(corollary_1_to_singletons_are_sets,axiom,
member(singleton(Y),unordered_pair(X,singleton(Y))) ).
%----(SS2): a set belongs to its singleton.
%----(u = x), (u e universal_class) --> (u e {x}).
cnf(set_in_its_singleton,axiom,
( ~ member(X,universal_class)
| member(X,singleton(X)) ) ).
%----corollary
cnf(corollary_to_set_in_its_singleton,axiom,
( ~ member(X,universal_class)
| singleton(X) != null_class ) ).
%----Not in the paper
cnf(null_class_in_its_singleton,axiom,
member(null_class,singleton(null_class)) ).
%----(SS3): only x can belong to {x}.
cnf(only_member_in_singleton,axiom,
( ~ member(Y,singleton(X))
| Y = X ) ).
%----(SS4): if x is not a set, {x} = O.
cnf(singleton_is_null_class,axiom,
( member(X,universal_class)
| singleton(X) = null_class ) ).
%----(SS5): a singleton set is determined by its element.
cnf(singleton_identified_by_element1,axiom,
( singleton(X) != singleton(Y)
| ~ member(X,universal_class)
| X = Y ) ).
cnf(singleton_identified_by_element2,axiom,
( singleton(X) != singleton(Y)
| ~ member(Y,universal_class)
| X = Y ) ).
%----(SS5.5).
%----Not in the paper
cnf(singleton_in_unordered_pair3,axiom,
( unordered_pair(Y,Z) != singleton(X)
| ~ member(X,universal_class)
| X = Y
| X = Z ) ).
%----(SS6): existence of memb.
%----a:x:e:u:(((u e universal_class) & x = {u}) | (-e:y:((y
%----e universal_class) & x = {y}) & u = x)).
cnf(member_exists1,axiom,
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ) ).
cnf(member_exists2,axiom,
( ~ member(Y,universal_class)
| singleton(member_of(singleton(Y))) = singleton(Y) ) ).
cnf(member_exists3,axiom,
( member(member_of(X),universal_class)
| member_of(X) = X ) ).
cnf(member_exists4,axiom,
( singleton(member_of(X)) = X
| member_of(X) = X ) ).
%----(SS7): uniqueness of memb of a singleton set.
%----a:x:a:u:(((u e universal_class) & x = {u}) ==> member_of(x) = u)
cnf(member_of_singleton_is_unique,axiom,
( ~ member(U,universal_class)
| member_of(singleton(U)) = U ) ).
cnf(prove_member_of_non_singleton_unique2_1,negated_conjecture,
singleton(member_of1(x)) != x ).
cnf(prove_member_of_non_singleton_unique2_2,negated_conjecture,
member_of(x) != x ).
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