SET007 Axioms: SET007+541.ax
%------------------------------------------------------------------------------
% File : SET007+541 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Tone Reflex of Topological Space
% 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_1topsp [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 40 ( 3 unt; 0 def)
% Number of atoms : 264 ( 21 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 282 ( 58 ~; 0 |; 133 &)
% ( 9 <=>; 82 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 0 con; 1-5 aty)
% Number of variables : 102 ( 95 !; 7 ?)
% 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(rc1_t_1topsp,axiom,
! [A] :
? [B] :
( m1_t_1topsp(B,A)
& v1_t_1topsp(B,A) ) ).
fof(rc2_t_1topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_eqrel_1(B,A)
& ~ v1_xboole_0(B)
& v1_setfam_1(B) ) ) ).
fof(rc3_t_1topsp,axiom,
! [A] :
? [B] :
( m1_t_1topsp(B,A)
& ~ v1_xboole_0(B)
& v1_t_1topsp(B,A) ) ).
fof(fc1_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k4_t_1topsp(A))
& v1_pre_topc(k4_t_1topsp(A))
& v2_pre_topc(k4_t_1topsp(A)) ) ) ).
fof(fc2_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k4_t_1topsp(A))
& v1_pre_topc(k4_t_1topsp(A))
& v2_pre_topc(k4_t_1topsp(A))
& v1_urysohn1(k4_t_1topsp(A)) ) ) ).
fof(t1_t_1topsp,axiom,
$true ).
fof(t2_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k16_borsuk_1(A,B))))
=> k5_pre_topc(A,k16_borsuk_1(A,B),k17_borsuk_1(A,B),C) = k3_tarski(C) ) ) ) ).
fof(t3_t_1topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> k4_xboole_0(k5_setfam_1(A,B),k3_tarski(C)) = k3_tarski(k4_xboole_0(B,C)) ) ) ) ).
fof(t4_t_1topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r2_hidden(B,C)
=> k3_tarski(k4_xboole_0(C,k1_tarski(B))) = k4_xboole_0(A,B) ) ) ) ) ).
fof(t5_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k16_borsuk_1(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( D = k3_tarski(C)
=> ( v4_pre_topc(C,k16_borsuk_1(A,B))
<=> v4_pre_topc(D,A) ) ) ) ) ) ) ).
fof(d1_t_1topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( D = k1_t_1topsp(A,B,C)
<=> ( r2_hidden(B,D)
& r2_hidden(D,C) ) ) ) ) ) ) ).
fof(t6_t_1topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( ! [D] :
( m1_subset_1(D,A)
=> k1_t_1topsp(A,D,B) = k1_t_1topsp(A,D,C) )
=> B = C ) ) ) ) ).
fof(t7_t_1topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m1_eqrel_1(k1_tarski(A),A) ) ).
fof(d2_t_1topsp,axiom,
! [A,B] :
( m1_t_1topsp(B,A)
<=> r1_tarski(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(d3_t_1topsp,axiom,
! [A,B] :
( m1_t_1topsp(B,A)
=> ( v1_t_1topsp(B,A)
<=> ! [C] :
( r2_hidden(C,B)
=> m1_eqrel_1(C,A) ) ) ) ).
fof(t8_t_1topsp,axiom,
! [A,B] :
( m1_eqrel_1(B,A)
=> ( v1_t_1topsp(k1_tarski(B),A)
& m1_t_1topsp(k1_tarski(B),A) ) ) ).
fof(t9_t_1topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ~ r1_xboole_0(k1_t_1topsp(A,C,B),k1_t_1topsp(A,D,B))
=> k1_t_1topsp(A,C,B) = k1_t_1topsp(A,D,B) ) ) ) ) ) ).
fof(t10_t_1topsp,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_eqrel_1(C,B)
=> ~ ( r2_hidden(A,C)
& ! [D] :
( m1_subset_1(D,B)
=> A != k1_t_1topsp(B,D,C) ) ) ) ) ).
fof(d6_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k4_t_1topsp(A) = k16_borsuk_1(A,k2_t_1topsp(u1_struct_0(A),k3_t_1topsp(A))) ) ).
fof(t12_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v1_urysohn1(k4_t_1topsp(A)) ) ).
fof(d7_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k5_t_1topsp(A) = k17_borsuk_1(A,k2_t_1topsp(u1_struct_0(A),k3_t_1topsp(A))) ) ).
fof(t14_t_1topsp,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(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_urysohn1(B)
=> ! [D,E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( D = k1_t_1topsp(u1_struct_0(A),E,k2_t_1topsp(u1_struct_0(A),k3_t_1topsp(A)))
=> r1_tarski(D,k5_pre_topc(A,B,C,k1_struct_0(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)))) ) ) ) ) ) ) ).
fof(t15_t_1topsp,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(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_urysohn1(B)
=> ! [D] :
~ ( r2_hidden(D,u1_struct_0(k4_t_1topsp(A)))
& ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ~ ( r2_hidden(E,k2_relat_1(C))
& r1_tarski(D,k5_pre_topc(A,B,C,k1_struct_0(B,E))) ) ) ) ) ) ) ) ).
fof(t16_t_1topsp,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(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ~ ( v1_urysohn1(B)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k4_t_1topsp(A)),u1_struct_0(B))
& v5_pre_topc(D,k4_t_1topsp(A),B)
& m2_relset_1(D,u1_struct_0(k4_t_1topsp(A)),u1_struct_0(B)) )
=> C != k4_borsuk_1(A,k4_t_1topsp(A),B,k5_t_1topsp(A),D) ) ) ) ) ) ).
fof(d8_t_1topsp,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(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(k4_t_1topsp(A)),u1_struct_0(k4_t_1topsp(B)))
& v5_pre_topc(D,k4_t_1topsp(A),k4_t_1topsp(B))
& m2_relset_1(D,u1_struct_0(k4_t_1topsp(A)),u1_struct_0(k4_t_1topsp(B))) )
=> ( D = k6_t_1topsp(A,B,C)
<=> k4_borsuk_1(A,B,k4_t_1topsp(B),C,k5_t_1topsp(B)) = k4_borsuk_1(A,k4_t_1topsp(A),k4_t_1topsp(B),k5_t_1topsp(A),D) ) ) ) ) ) ).
fof(dt_m1_t_1topsp,axiom,
$true ).
fof(existence_m1_t_1topsp,axiom,
! [A] :
? [B] : m1_t_1topsp(B,A) ).
fof(dt_k1_t_1topsp,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_eqrel_1(C,A) )
=> m1_subset_1(k1_t_1topsp(A,B,C),k1_zfmisc_1(A)) ) ).
fof(dt_k2_t_1topsp,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_t_1topsp(B,A)
& m1_t_1topsp(B,A) )
=> ( ~ v1_xboole_0(k2_t_1topsp(A,B))
& m1_eqrel_1(k2_t_1topsp(A,B),A) ) ) ).
fof(dt_k3_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v1_xboole_0(k3_t_1topsp(A))
& v1_t_1topsp(k3_t_1topsp(A),u1_struct_0(A))
& m1_t_1topsp(k3_t_1topsp(A),u1_struct_0(A)) ) ) ).
fof(dt_k4_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_pre_topc(k4_t_1topsp(A))
& l1_pre_topc(k4_t_1topsp(A)) ) ) ).
fof(dt_k5_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k5_t_1topsp(A))
& v1_funct_2(k5_t_1topsp(A),u1_struct_0(A),u1_struct_0(k4_t_1topsp(A)))
& v5_pre_topc(k5_t_1topsp(A),A,k4_t_1topsp(A))
& m2_relset_1(k5_t_1topsp(A),u1_struct_0(A),u1_struct_0(k4_t_1topsp(A))) ) ) ).
fof(dt_k6_t_1topsp,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_funct_1(k6_t_1topsp(A,B,C))
& v1_funct_2(k6_t_1topsp(A,B,C),u1_struct_0(k4_t_1topsp(A)),u1_struct_0(k4_t_1topsp(B)))
& v5_pre_topc(k6_t_1topsp(A,B,C),k4_t_1topsp(A),k4_t_1topsp(B))
& m2_relset_1(k6_t_1topsp(A,B,C),u1_struct_0(k4_t_1topsp(A)),u1_struct_0(k4_t_1topsp(B))) ) ) ).
fof(d4_t_1topsp,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_t_1topsp(B,A)
& m1_t_1topsp(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,A) )
=> ( C = k2_t_1topsp(A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> k1_t_1topsp(A,D,C) = k1_setfam_1(a_3_0_t_1topsp(A,B,D)) ) ) ) ) ) ).
fof(t11_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_t_1topsp(a_1_0_t_1topsp(A),u1_struct_0(A))
& m1_t_1topsp(a_1_0_t_1topsp(A),u1_struct_0(A)) ) ) ).
fof(d5_t_1topsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k3_t_1topsp(A) = a_1_0_t_1topsp(A) ) ).
fof(t13_t_1topsp,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(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_urysohn1(B)
=> ( m1_eqrel_1(a_3_1_t_1topsp(A,B,C),u1_struct_0(A))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(D,a_3_1_t_1topsp(A,B,C))
=> v4_pre_topc(D,A) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_t_1topsp,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_t_1topsp(C,B)
& m1_t_1topsp(C,B)
& m1_subset_1(D,B) )
=> ( r2_hidden(A,a_3_0_t_1topsp(B,C,D))
<=> ? [E] :
( m1_eqrel_1(E,B)
& A = k1_t_1topsp(B,D,E)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_1_0_t_1topsp,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( r2_hidden(A,a_1_0_t_1topsp(B))
<=> ? [C] :
( m1_eqrel_1(C,u1_struct_0(B))
& A = C
& v2_tops_2(C,B) ) ) ) ).
fof(fraenkel_a_3_1_t_1topsp,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C)
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( r2_hidden(A,a_3_1_t_1topsp(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(C))
& A = k5_pre_topc(B,C,D,k1_struct_0(C,E))
& r2_hidden(E,k2_relat_1(D)) ) ) ) ).
%------------------------------------------------------------------------------