%--------------------------------------------------------------------------
% File     : SET085-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% Domain   : Set Theory
% Problem  : Unordered pair that is a singleton
% Version  : [Qua92] axioms : Augmented.
% English  :

% Refs     : [Qua92] Quaife (1992), Automated Deduction in von Neumann-Bern
% Source   : [Quaife]
% Names    : SS5.5 [Quaife]

% Status   : Unsatisfiable
% Rating   : 0.10 v9.0.0, 0.15 v8.2.0, 0.14 v8.1.0, 0.05 v7.5.0, 0.11 v7.4.0, 0.12 v7.3.0, 0.08 v7.1.0, 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, 0.25 v5.4.0, 0.30 v5.3.0, 0.22 v5.2.0, 0.19 v5.1.0, 0.18 v5.0.0, 0.14 v4.1.0, 0.15 v4.0.1, 0.18 v3.7.0, 0.20 v3.5.0, 0.18 v3.4.0, 0.17 v3.3.0, 0.14 v3.2.0, 0.15 v3.1.0, 0.27 v2.7.0, 0.17 v2.6.0, 0.11 v2.5.0, 0.27 v2.4.0, 0.12 v2.3.0, 0.25 v2.2.1, 0.33 v2.2.0, 0.33 v2.1.0
% Syntax   : Number of clauses     :  132 (  44 unt;  16 nHn;  90 RR)
%            Number of literals    :  259 (  66 equ; 119 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;  11 con; 0-3 aty)
%            Number of variables   :  251 (  46 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 ) ).

%----(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 ) ).

cnf(prove_singleton_in_unordered_pair3_1,negated_conjecture,
    unordered_pair(y,z) = singleton(x) ).

cnf(prove_singleton_in_unordered_pair3_2,negated_conjecture,
    member(x,universal_class) ).

cnf(prove_singleton_in_unordered_pair3_3,negated_conjecture,
    x != y ).

cnf(prove_singleton_in_unordered_pair3_4,negated_conjecture,
    x != z ).

%--------------------------------------------------------------------------