%--------------------------------------------------------------------------
% File     : TOP004-2 : TPTP v9.0.0. Released v1.0.0.
% Domain   : Topology
% Problem  : Topology generated by a basis forms a topological space, part 4
% Version  : [WM89] axioms : Incomplete > Reduced & Augmented > Incomplete.
% English  :

% Refs     : [WM89]  Wick & McCune (1989), Automated Reasoning about Elemen
% Source   : [WM89]
% Names    : Lemma 1d [WM89]

% Status   : Unsatisfiable
% Rating   : 0.00 v6.3.0, 0.14 v6.2.0, 0.00 v2.0.0
% Syntax   : Number of clauses     :   21 (   5 unt;   1 nHn;  17 RR)
%            Number of literals    :   59 (   0 equ;  38 neg)
%            Maximal clause size   :    6 (   2 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    5 (   5 usr;   0 prp; 2-2 aty)
%            Number of functors    :    8 (   8 usr;   2 con; 0-5 aty)
%            Number of variables   :   59 (   6 sgn)
% SPC      : CNF_UNS_RFO_NEQ_NHN

% Comments : The axioms in this version are known to be incomplete. To
%            make them complete it is be necessary to add appropriate set
%            theory axioms.
%--------------------------------------------------------------------------
%----Include Point-set topology axioms
% include('Axioms/TOP001-0.ax').
%--------------------------------------------------------------------------
%----Sigma (union of members).
cnf(union_of_members_3,axiom,
    ( element_of_set(U,union_of_members(Vf))
    | ~ element_of_set(U,Uu1)
    | ~ element_of_collection(Uu1,Vf) ) ).

%----Basis for a topology
cnf(basis_for_topology_28,axiom,
    ( ~ basis(X,Vf)
    | equal_sets(union_of_members(Vf),X) ) ).

cnf(basis_for_topology_29,axiom,
    ( ~ basis(X,Vf)
    | ~ element_of_set(Y,X)
    | ~ element_of_collection(Vb1,Vf)
    | ~ element_of_collection(Vb2,Vf)
    | ~ element_of_set(Y,intersection_of_sets(Vb1,Vb2))
    | element_of_set(Y,f6(X,Vf,Y,Vb1,Vb2)) ) ).

cnf(basis_for_topology_30,axiom,
    ( ~ basis(X,Vf)
    | ~ element_of_set(Y,X)
    | ~ element_of_collection(Vb1,Vf)
    | ~ element_of_collection(Vb2,Vf)
    | ~ element_of_set(Y,intersection_of_sets(Vb1,Vb2))
    | element_of_collection(f6(X,Vf,Y,Vb1,Vb2),Vf) ) ).

cnf(basis_for_topology_31,axiom,
    ( ~ basis(X,Vf)
    | ~ element_of_set(Y,X)
    | ~ element_of_collection(Vb1,Vf)
    | ~ element_of_collection(Vb2,Vf)
    | ~ element_of_set(Y,intersection_of_sets(Vb1,Vb2))
    | subset_sets(f6(X,Vf,Y,Vb1,Vb2),intersection_of_sets(Vb1,Vb2)) ) ).

%----Topology generated by a basis
cnf(topology_generated_37,axiom,
    ( ~ element_of_collection(U,top_of_basis(Vf))
    | ~ element_of_set(X,U)
    | element_of_set(X,f10(Vf,U,X)) ) ).

cnf(topology_generated_38,axiom,
    ( ~ element_of_collection(U,top_of_basis(Vf))
    | ~ element_of_set(X,U)
    | element_of_collection(f10(Vf,U,X),Vf) ) ).

cnf(topology_generated_39,axiom,
    ( ~ element_of_collection(U,top_of_basis(Vf))
    | ~ element_of_set(X,U)
    | subset_sets(f10(Vf,U,X),U) ) ).

cnf(topology_generated_40,axiom,
    ( element_of_collection(U,top_of_basis(Vf))
    | element_of_set(f11(Vf,U),U) ) ).

cnf(topology_generated_41,axiom,
    ( element_of_collection(U,top_of_basis(Vf))
    | ~ element_of_set(f11(Vf,U),Uu11)
    | ~ element_of_collection(Uu11,Vf)
    | ~ subset_sets(Uu11,U) ) ).

cnf(set_theory_12,axiom,
    ( ~ subset_sets(X,Y)
    | ~ subset_sets(Y,Z)
    | subset_sets(X,Z) ) ).

cnf(set_theory_13,axiom,
    ( ~ element_of_set(Z,intersection_of_sets(X,Y))
    | element_of_set(Z,X) ) ).

cnf(set_theory_14,axiom,
    ( ~ element_of_set(Z,intersection_of_sets(X,Y))
    | element_of_set(Z,Y) ) ).

cnf(set_theory_15,axiom,
    ( element_of_set(Z,intersection_of_sets(X,Y))
    | ~ element_of_set(Z,X)
    | ~ element_of_set(Z,Y) ) ).

cnf(set_theory_16,axiom,
    ( ~ subset_sets(X,Y)
    | ~ subset_sets(U,V)
    | subset_sets(intersection_of_sets(X,U),intersection_of_sets(Y,V)) ) ).

cnf(set_theory_17,axiom,
    ( ~ equal_sets(X,Y)
    | ~ element_of_set(Z,X)
    | element_of_set(Z,Y) ) ).

cnf(set_theory_18,axiom,
    equal_sets(intersection_of_sets(X,Y),intersection_of_sets(Y,X)) ).

cnf(lemma_1d_1,negated_conjecture,
    basis(cx,f) ).

cnf(lemma_1d_2,negated_conjecture,
    element_of_collection(U,top_of_basis(f)) ).

cnf(lemma_1d_3,negated_conjecture,
    element_of_collection(V,top_of_basis(f)) ).

cnf(lemma_1d_4,negated_conjecture,
    ~ element_of_collection(intersection_of_sets(U,V),top_of_basis(f)) ).

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