SET007 Axioms: SET007+208.ax
%------------------------------------------------------------------------------
% File : SET007+208 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Connected 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 : connsp_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 55 ( 1 unt; 0 def)
% Number of atoms : 432 ( 44 equ)
% Maximal formula atoms : 14 ( 7 avg)
% Number of connectives : 428 ( 51 ~; 4 |; 156 &)
% ( 19 <=>; 198 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 16 ( 16 usr; 1 con; 0-4 aty)
% Number of variables : 174 ( 168 !; 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(d1_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_connsp_1(A,B,C)
<=> ( r1_xboole_0(k6_pre_topc(A,B),C)
& r1_xboole_0(B,k6_pre_topc(A,C)) ) ) ) ) ) ).
fof(t1_connsp_1,axiom,
$true ).
fof(t2_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_connsp_1(A,B,C)
=> r1_xboole_0(B,C) ) ) ) ) ).
fof(t3_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C)
& v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& r1_xboole_0(B,C) )
=> r1_connsp_1(A,B,C) ) ) ) ) ).
fof(t4_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C)
& v3_pre_topc(B,A)
& v3_pre_topc(C,A)
& r1_xboole_0(B,C) )
=> r1_connsp_1(A,B,C) ) ) ) ) ).
fof(t5_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C)
& r1_connsp_1(A,B,C) )
=> ( v3_pre_topc(B,A)
& v4_pre_topc(B,A)
& v3_pre_topc(C,A)
& v4_pre_topc(C,A) ) ) ) ) ) ).
fof(t6_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( E = C
& F = D
& r1_connsp_1(B,E,F) )
=> r1_connsp_1(A,C,D) ) ) ) ) ) ) ) ).
fof(t7_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( C = E
& D = F
& r1_tarski(k4_subset_1(u1_struct_0(A),C,D),k2_pre_topc(B))
& r1_connsp_1(A,C,D) )
=> r1_connsp_1(B,E,F) ) ) ) ) ) ) ) ).
fof(t8_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_connsp_1(A,B,C)
& r1_tarski(D,B)
& r1_tarski(E,C) )
=> r1_connsp_1(A,D,E) ) ) ) ) ) ) ).
fof(t9_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_connsp_1(A,B,C)
& r1_connsp_1(A,B,D) )
=> r1_connsp_1(A,B,k4_subset_1(u1_struct_0(A),C,D)) ) ) ) ) ) ).
fof(t10_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ( v4_pre_topc(B,A)
& v4_pre_topc(C,A) )
| ( v3_pre_topc(B,A)
& v3_pre_topc(C,A) ) )
=> r1_connsp_1(A,k6_subset_1(u1_struct_0(A),B,C),k6_subset_1(u1_struct_0(A),C,B)) ) ) ) ) ).
fof(d2_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v1_connsp_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C)
& r1_connsp_1(A,B,C)
& B != k1_pre_topc(A)
& C != k1_pre_topc(A) ) ) ) ) ) ).
fof(t11_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C)
& B != k1_pre_topc(A)
& C != k1_pre_topc(A)
& v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& r1_xboole_0(B,C) ) ) ) ) ) ).
fof(t12_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C)
& B != k1_pre_topc(A)
& C != k1_pre_topc(A)
& v3_pre_topc(B,A)
& v3_pre_topc(C,A)
& r1_xboole_0(B,C) ) ) ) ) ) ).
fof(t13_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B != k1_pre_topc(A)
& B != k2_pre_topc(A)
& r1_xboole_0(k6_pre_topc(A,B),k6_pre_topc(A,k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B))) ) ) ) ) ).
fof(t14_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(B,A)
& v4_pre_topc(B,A)
& B != k1_pre_topc(A)
& B != k2_pre_topc(A) ) ) ) ) ).
fof(t15_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [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))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v5_pre_topc(C,A,B)
& k4_pre_topc(A,B,C,k2_pre_topc(A)) = k2_pre_topc(B)
& v1_connsp_1(A) )
=> v1_connsp_1(B) ) ) ) ) ).
fof(d3_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_connsp_1(B,A)
<=> v1_connsp_1(k3_pre_topc(A,B)) ) ) ) ).
fof(t16_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_connsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B = k4_subset_1(u1_struct_0(A),C,D)
& r1_connsp_1(A,C,D)
& C != k1_pre_topc(A)
& D != k1_pre_topc(A) ) ) ) ) ) ) ).
fof(t17_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v2_connsp_1(B,A)
& r1_tarski(B,k4_subset_1(u1_struct_0(A),C,D))
& r1_connsp_1(A,C,D)
& ~ r1_tarski(B,C)
& ~ r1_tarski(B,D) ) ) ) ) ) ).
fof(t18_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(C,A) )
=> ( r1_connsp_1(A,B,C)
| v2_connsp_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ) ).
fof(t19_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& r1_tarski(B,C)
& r1_tarski(C,k6_pre_topc(A,B)) )
=> v2_connsp_1(C,A) ) ) ) ) ).
fof(t20_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_connsp_1(B,A)
=> v2_connsp_1(k6_pre_topc(A,B),A) ) ) ) ).
fof(t21_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_connsp_1(A)
& v2_connsp_1(B,A)
& k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) = k4_subset_1(u1_struct_0(A),C,D)
& r1_connsp_1(A,C,D) )
=> ( v2_connsp_1(k4_subset_1(u1_struct_0(A),B,C),A)
& v2_connsp_1(k4_subset_1(u1_struct_0(A),B,D),A) ) ) ) ) ) ) ).
fof(t22_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) = k4_subset_1(u1_struct_0(A),C,D)
& r1_connsp_1(A,C,D)
& v4_pre_topc(B,A) )
=> ( v4_pre_topc(k4_subset_1(u1_struct_0(A),B,C),A)
& v4_pre_topc(k4_subset_1(u1_struct_0(A),B,D),A) ) ) ) ) ) ) ).
fof(t23_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v2_connsp_1(B,A)
& ~ r1_xboole_0(B,C)
& k6_subset_1(u1_struct_0(A),B,C) != k1_pre_topc(A)
& r1_xboole_0(B,k2_tops_1(A,C)) ) ) ) ) ).
fof(t24_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = D
=> ( v2_connsp_1(C,A)
<=> v2_connsp_1(D,B) ) ) ) ) ) ) ).
fof(t25_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& v2_connsp_1(k4_subset_1(u1_struct_0(A),B,C),A)
& v2_connsp_1(k5_subset_1(u1_struct_0(A),B,C),A) )
=> ( v2_connsp_1(B,A)
& v2_connsp_1(C,A) ) ) ) ) ) ).
fof(t26_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v2_connsp_1(C,A) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( C != k1_pre_topc(A)
& r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(D,B)
& D != C
& r1_connsp_1(A,C,D) ) ) ) )
| v2_connsp_1(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ) ).
fof(t27_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v2_connsp_1(C,A) ) )
=> ( k6_setfam_1(u1_struct_0(A),B) = k1_pre_topc(A)
| v2_connsp_1(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ) ).
fof(t28_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_connsp_1(k2_pre_topc(A),A)
<=> v1_connsp_1(A) ) ) ).
fof(t29_connsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v2_connsp_1(k1_struct_0(A,B),A) ) ) ).
fof(d4_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_connsp_1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& v2_connsp_1(D,A)
& r2_hidden(B,D)
& r2_hidden(C,D) ) ) ) ) ) ).
fof(t30_connsp_1,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))
=> r2_connsp_1(A,B,C) ) )
=> v1_connsp_1(A) ) ) ).
fof(t31_connsp_1,axiom,
! [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))
=> r2_connsp_1(A,B,C) ) )
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_connsp_1(A,B,C) ) ) ) ) ).
fof(t32_connsp_1,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))
=> r2_connsp_1(A,B,C) ) )
=> v1_connsp_1(A) ) ) ).
fof(t33_connsp_1,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,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(D,C)
<=> ( v2_connsp_1(D,A)
& r2_hidden(B,D) ) ) )
& C = k1_xboole_0 ) ) ) ) ).
fof(d5_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( r3_connsp_1(A,B)
<=> ( v2_connsp_1(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(C,A)
& r1_tarski(B,C) )
=> B = C ) ) ) ) ) ) ).
fof(t34_connsp_1,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)))
=> ~ ( r3_connsp_1(A,B)
& B = k1_pre_topc(A) ) ) ) ).
fof(t35_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( r3_connsp_1(A,B)
=> v4_pre_topc(B,A) ) ) ) ).
fof(t36_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r3_connsp_1(A,B)
& r3_connsp_1(A,C)
& B != C
& ~ r1_connsp_1(A,B,C) ) ) ) ) ).
fof(t37_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r3_connsp_1(A,B)
& r3_connsp_1(A,C)
& B != C
& ~ r1_xboole_0(B,C) ) ) ) ) ).
fof(t38_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_connsp_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r3_connsp_1(A,C)
& ~ r1_xboole_0(B,C)
& ~ r1_tarski(B,C) ) ) ) ) ) ).
fof(d6_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_connsp_1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
& D = C
& r3_connsp_1(k3_pre_topc(A,B),D) ) ) ) ) ) ).
fof(t39_connsp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_connsp_1(A)
& v2_connsp_1(B,A)
& r4_connsp_1(A,k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B),C) )
=> v2_connsp_1(k6_subset_1(u1_struct_0(A),k2_pre_topc(A),C),A) ) ) ) ) ).
fof(d7_connsp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = k1_connsp_1(A,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,D)
<=> ( v2_connsp_1(E,A)
& r2_hidden(B,E) ) ) )
& k5_setfam_1(u1_struct_0(A),D) = C ) ) ) ) ) ).
fof(t40_connsp_1,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,k1_connsp_1(A,B)) ) ) ).
fof(t41_connsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v2_connsp_1(k1_connsp_1(A,B),A) ) ) ).
fof(t42_connsp_1,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)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( v2_connsp_1(B,A)
& r1_tarski(k1_connsp_1(A,C),B) )
=> B = k1_connsp_1(A,C) ) ) ) ) ).
fof(t43_connsp_1,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)))
=> ( r3_connsp_1(A,B)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B = k1_connsp_1(A,C) ) ) ) ) ).
fof(t44_connsp_1,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)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( r3_connsp_1(A,B)
& r2_hidden(C,B) )
=> B = k1_connsp_1(A,C) ) ) ) ) ).
fof(t45_connsp_1,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)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( B = k1_connsp_1(A,C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> k1_connsp_1(A,D) = B ) ) ) ) ) ) ).
fof(t46_connsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
<=> r3_connsp_1(A,C) ) )
=> r1_pre_topc(A,B) ) ) ) ).
fof(symmetry_r1_connsp_1,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_connsp_1(A,B,C)
=> r1_connsp_1(A,C,B) ) ) ).
fof(dt_k1_connsp_1,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_connsp_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
%------------------------------------------------------------------------------