SET007 Axioms: SET007+399.ax
%------------------------------------------------------------------------------
% File : SET007+399 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Tzero Topological Spaces
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t_0topsp [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 31 ( 0 unt; 0 def)
% Number of atoms : 237 ( 19 equ)
% Maximal formula atoms : 16 ( 7 avg)
% Number of connectives : 248 ( 42 ~; 3 |; 110 &)
% ( 11 <=>; 82 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 22 usr; 0 prp; 1-3 aty)
% Number of functors : 24 ( 24 usr; 0 con; 1-5 aty)
% Number of variables : 79 ( 73 !; 6 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k3_t_0topsp(A))
& v2_pre_topc(k3_t_0topsp(A)) ) ) ).
fof(rc1_t_0topsp,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_t_0topsp(A) ) ).
fof(t1_t_0topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,B)
=> ! [D] :
( m2_relset_1(D,A,B)
=> ( ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> ( r2_hidden(k4_tarski(E,F),C)
<=> r2_hidden(k4_tarski(E,F),D) ) ) )
=> C = D ) ) ) ) ) ).
fof(t2_t_0topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ( ( r2_hidden(E,D)
& k8_funct_2(A,B,C,E) = k8_funct_2(A,B,C,F) )
=> r2_hidden(F,D) ) ) )
=> k3_funct_2(A,B,C,k2_funct_2(A,B,C,D)) = D ) ) ) ) ) ).
fof(d1_t_0topsp,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ( r1_t_0topsp(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& v3_tops_2(C,A,B) ) ) ) ) ).
fof(d2_t_0topsp,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_t_0topsp(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(D,A)
=> v3_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) ).
fof(d3_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( B = k1_t_0topsp(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(k4_tarski(C,D),B)
<=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(E,A)
=> ( r2_hidden(C,E)
<=> r2_hidden(D,E) ) ) ) ) ) ) ) ) ) ).
fof(d4_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> k2_t_0topsp(A) = k8_eqrel_1(u1_struct_0(A),k1_t_0topsp(A)) ) ).
fof(d5_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k3_t_0topsp(A) = k16_borsuk_1(A,k2_t_0topsp(A)) ) ).
fof(d6_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k4_t_0topsp(A) = k17_borsuk_1(A,k2_t_0topsp(A)) ) ).
fof(t3_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_hidden(B,k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),B)) ) ) ).
fof(t4_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( k1_relat_1(k4_t_0topsp(A)) = u1_struct_0(A)
& r1_tarski(k2_relat_1(k4_t_0topsp(A)),u1_struct_0(k3_t_0topsp(A))) ) ) ).
fof(t6_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_t_0topsp(A))))
=> ( v3_pre_topc(B,k3_t_0topsp(A))
<=> r2_hidden(k3_tarski(B),u1_pre_topc(A)) ) ) ) ).
fof(t7_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_t_0topsp(A)))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B = k6_eqrel_1(u1_struct_0(A),k1_t_0topsp(A),C) ) ) ) ).
fof(t8_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),B) = k6_eqrel_1(u1_struct_0(A),k1_t_0topsp(A),B) ) ) ).
fof(t9_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),C) = k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),B)
<=> r2_hidden(k8_borsuk_1(A,A,C,B),k1_t_0topsp(A)) ) ) ) ) ).
fof(t10_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),C) = k8_funct_2(u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(A),D) )
=> r2_hidden(D,B) ) ) ) ) ) ) ).
fof(t11_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,k2_t_0topsp(A))
=> ( r1_xboole_0(C,B)
| r1_tarski(C,B) ) ) ) ) ) ) ).
fof(t12_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v1_t_0topsp(k4_t_0topsp(A),A,k3_t_0topsp(A)) ) ).
fof(d7_t_0topsp,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v2_t_0topsp(A)
<=> ( v3_struct_0(A)
| ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& ( ( r2_hidden(B,D)
& ~ r2_hidden(C,D) )
| ( r2_hidden(C,D)
& ~ r2_hidden(B,D) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k3_t_0topsp(A))
& v2_pre_topc(k3_t_0topsp(A))
& v2_t_0topsp(k3_t_0topsp(A))
& l1_pre_topc(k3_t_0topsp(A)) ) ) ).
fof(t14_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k3_t_0topsp(B)),u1_struct_0(k3_t_0topsp(A)))
& m2_relset_1(C,u1_struct_0(k3_t_0topsp(B)),u1_struct_0(k3_t_0topsp(A)))
& v3_tops_2(C,k3_t_0topsp(B),k3_t_0topsp(A))
& r1_rfinseq(k4_t_0topsp(A),k7_funct_2(u1_struct_0(B),u1_struct_0(k3_t_0topsp(B)),u1_struct_0(k3_t_0topsp(A)),k4_t_0topsp(B),C)) )
=> r1_t_0topsp(A,B) ) ) ) ).
fof(t15_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(k8_borsuk_1(A,A,D,E),k1_t_0topsp(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D) = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) ) ) ) ) ) ) ).
fof(t16_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k4_pre_topc(A,B,C,k6_eqrel_1(u1_struct_0(A),k1_t_0topsp(A),D)) = k1_struct_0(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D)) ) ) ) ) ).
fof(t17_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k3_t_0topsp(A)),u1_struct_0(B))
& v5_pre_topc(D,k3_t_0topsp(A),B)
& m2_relset_1(D,u1_struct_0(k3_t_0topsp(A)),u1_struct_0(B))
& C = k4_borsuk_1(A,k3_t_0topsp(A),B,k4_t_0topsp(A),D) ) ) ) ) ).
fof(dt_k1_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v3_relat_2(k1_t_0topsp(A))
& v8_relat_2(k1_t_0topsp(A))
& v1_partfun1(k1_t_0topsp(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k1_t_0topsp(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k2_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( ~ v1_xboole_0(k2_t_0topsp(A))
& m1_eqrel_1(k2_t_0topsp(A),u1_struct_0(A)) ) ) ).
fof(dt_k3_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_pre_topc(k3_t_0topsp(A))
& l1_pre_topc(k3_t_0topsp(A)) ) ) ).
fof(dt_k4_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k4_t_0topsp(A))
& v1_funct_2(k4_t_0topsp(A),u1_struct_0(A),u1_struct_0(k3_t_0topsp(A)))
& v5_pre_topc(k4_t_0topsp(A),A,k3_t_0topsp(A))
& m2_relset_1(k4_t_0topsp(A),u1_struct_0(A),u1_struct_0(k3_t_0topsp(A))) ) ) ).
fof(t5_t_0topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( u1_struct_0(k3_t_0topsp(A)) = k2_t_0topsp(A)
& u1_pre_topc(k3_t_0topsp(A)) = a_1_0_t_0topsp(A) ) ) ).
fof(fraenkel_a_1_0_t_0topsp,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( r2_hidden(A,a_1_0_t_0topsp(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k2_t_0topsp(B)))
& A = C
& r2_hidden(k3_tarski(C),u1_pre_topc(B)) ) ) ) ).
%------------------------------------------------------------------------------