TPTP Axioms File: SET004-1.ax


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% File     : SET004-1 : TPTP v7.5.0. Bugfixed v1.0.1.
% Domain   : Set Theory (Boolean Algebra definitions)
% Axioms   : Set theory (Boolean algebra) axioms based on NBG set theory
% Version  : [Qua92a] axioms.
% English  :

% Refs     : [Qua92a] Quaife (1992), Automated Deduction in von Neumann-Bern
%          : [Qua92b] Quaife (1992), Email to G. Sutcliffe
% Source   : [Qua92b]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses    :   21 (   0 non-Horn;   8 unit;  17 RR)
%            Number of atoms      :   37 (  10 equality)
%            Maximal clause size  :    3 (   2 average)
%            Number of predicates :    5 (   0 propositional; 1-3 arity)
%            Number of functors   :   26 (   7 constant; 0-3 arity)
%            Number of variables  :   38 (   7 singleton)
%            Maximal term depth   :    5 (   2 average)
% SPC      : 

% Comments : Requires SET004-0.ax
%          : Not all of these definitions appear in [Qua92a]. Some were
%            extracted from [Qua92b]
% Bugfixes : v1.0.1 - Duplicate axioms single_valued_term_defn? removed.
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%----(CO25DEF): Definition of compose_class(x) term, where x may
%----be a class. Not in [Quaife, 1992].
cnf(compose_class_definition1,axiom,
    ( subclass(compose_class(X),cross_product(universal_class,universal_class)) )).

cnf(compose_class_definition2,axiom,
    ( ~ member(ordered_pair(Y,Z),compose_class(X))
    | compose(X,Y) = Z )).

cnf(compose_class_definition3,axiom,
    ( ~ member(ordered_pair(Y,Z),cross_product(universal_class,universal_class))
    | compose(X,Y) != Z
    | member(ordered_pair(Y,Z),compose_class(X)) )).

%----(CO20DEF): Definition of composition_function. Not in [Quaife,
%----1992].
cnf(definition_of_composition_function1,axiom,
    ( subclass(composition_function,cross_product(universal_class,cross_product(universal_class,universal_class))) )).

cnf(definition_of_composition_function2,axiom,
    ( ~ member(ordered_pair(X,ordered_pair(Y,Z)),composition_function)
    | compose(X,Y) = Z )).

cnf(definition_of_composition_function3,axiom,
    ( ~ member(ordered_pair(X,Y),cross_product(universal_class,universal_class))
    | member(ordered_pair(X,ordered_pair(Y,compose(X,Y))),composition_function) )).

%----(DODEF11): Definition of domain_relation by the class existence
%----theorem. Not in [Quaife, 19992].
cnf(definition_of_domain_relation1,axiom,
    ( subclass(domain_relation,cross_product(universal_class,universal_class)) )).

cnf(definition_of_domain_relation2,axiom,
    ( ~ member(ordered_pair(X,Y),domain_relation)
    | domain_of(X) = Y )).

cnf(definition_of_domain_relation3,axiom,
    ( ~ member(X,universal_class)
    | member(ordered_pair(X,domain_of(X)),domain_relation) )).

%----(SV2DEF) Definitions of terms for (SV3) Called FU2DEF in Quaife's
%----email
cnf(single_valued_term_defn1,axiom,
    ( first(not_subclass_element(compose(X,inverse(X)),identity_relation)) = single_valued1(X) )).

cnf(single_valued_term_defn2,axiom,
    ( second(not_subclass_element(compose(X,inverse(X)),identity_relation)) = single_valued2(X) )).

cnf(single_valued_term_defn3,axiom,
    ( domain(X,image(inverse(X),singleton(single_valued1(X))),single_valued2(X)) = single_valued3(X) )).

%----(CO14DEF): Definition of singleton relation.
cnf(compose_can_define_singleton,axiom,
    ( intersection(complement(compose(element_relation,complement(identity_relation))),element_relation) = singleton_relation )).

%----(AP15): definition of application function. Not in [Qua92]
cnf(application_function_defn1,axiom,
    ( subclass(application_function,cross_product(universal_class,cross_product(universal_class,universal_class))) )).

cnf(application_function_defn2,axiom,
    ( ~ member(ordered_pair(X,ordered_pair(Y,Z)),application_function)
    | member(Y,domain_of(X)) )).

cnf(application_function_defn3,axiom,
    ( ~ member(ordered_pair(X,ordered_pair(Y,Z)),application_function)
    | apply(X,Y) = Z )).

cnf(application_function_defn4,axiom,
    ( ~ member(ordered_pair(X,ordered_pair(Y,Z)),cross_product(universal_class,cross_product(universal_class,universal_class)))
    | ~ member(Y,domain_of(X))
    | member(ordered_pair(X,ordered_pair(Y,apply(X,Y))),application_function) )).

%----Definition of maps. Not in [Qua92].
%----a:xf:a:x:a:y:(maps(xf,x,y) <=> function(xf) & domain(xf)
%----= x & range(xf) < y).
cnf(maps1,axiom,
    ( ~ maps(Xf,X,Y)
    | function(Xf) )).

cnf(maps2,axiom,
    ( ~ maps(Xf,X,Y)
    | domain_of(Xf) = X )).

cnf(maps3,axiom,
    ( ~ maps(Xf,X,Y)
    | subclass(range_of(Xf),Y) )).

cnf(maps4,axiom,
    ( ~ function(Xf)
    | ~ subclass(range_of(Xf),Y)
    | maps(Xf,domain_of(Xf),Y) )).

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