SET007 Axioms: SET007+451.ax
%------------------------------------------------------------------------------
% File : SET007+451 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Components and Unions of Components
% 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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 50 ( 0 unt; 0 def)
% Number of atoms : 337 ( 38 equ)
% Maximal formula atoms : 12 ( 6 avg)
% Number of connectives : 321 ( 34 ~; 10 |; 109 &)
% ( 5 <=>; 163 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-2 aty)
% Number of functors : 18 ( 18 usr; 1 con; 0-3 aty)
% Number of variables : 142 ( 137 !; 5 ?)
% 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_3,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)))
=> ( C = k1_connsp_3(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)
& r1_tarski(B,E) ) ) )
& k5_setfam_1(u1_struct_0(A),D) = C ) ) ) ) ) ).
fof(t1_connsp_3,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(C,A)
& r1_tarski(B,C) )
=> r1_tarski(B,k1_connsp_3(A,B)) ) ) ) ).
fof(t2_connsp_3,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(C,A)
& r1_tarski(B,C) ) )
=> k1_connsp_3(A,B) = k1_xboole_0 ) ) ) ).
fof(t3_connsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k1_connsp_3(A,k1_pre_topc(A)) = u1_struct_0(A) ) ).
fof(t4_connsp_3,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)))
=> ~ ( v2_connsp_1(B,A)
& k1_connsp_3(A,B) = k1_xboole_0 ) ) ) ).
fof(t5_connsp_3,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)
=> ( B = k1_xboole_0
| v2_connsp_1(k1_connsp_3(A,B),A) ) ) ) ) ).
fof(t6_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(C,A)
& r1_tarski(k1_connsp_3(A,B),C) )
=> C = k1_connsp_3(A,B) ) ) ) ) ).
fof(t7_connsp_3,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)
=> k1_connsp_3(A,B) = B ) ) ) ).
fof(t8_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
& v2_connsp_1(C,A)
& C != k1_xboole_0
& B = k1_connsp_3(A,C) ) ) ) ) ).
fof(t9_connsp_3,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)))
=> ( v2_connsp_1(B,A)
=> ( B = k1_xboole_0
| r3_connsp_1(A,k1_connsp_3(A,B)) ) ) ) ) ).
fof(t10_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r3_connsp_1(A,B)
& v2_connsp_1(C,A)
& r1_tarski(C,B) )
=> ( C = k1_xboole_0
| B = k1_connsp_3(A,C) ) ) ) ) ) ).
fof(t11_connsp_3,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)))
=> ( v2_connsp_1(B,A)
=> ( B = k1_xboole_0
| k1_connsp_3(A,k1_connsp_3(A,B)) = k1_connsp_3(A,B) ) ) ) ) ).
fof(t12_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(C,A)
& r1_tarski(B,C) )
=> ( B = k1_xboole_0
| k1_connsp_3(A,B) = k1_connsp_3(A,C) ) ) ) ) ) ).
fof(t13_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(C,A)
& r1_tarski(B,C) )
=> ( B = k1_xboole_0
| r1_tarski(C,k1_connsp_3(A,B)) ) ) ) ) ) ).
fof(t14_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(k4_subset_1(u1_struct_0(A),B,C),A) )
=> ( B = k1_xboole_0
| r1_tarski(k4_subset_1(u1_struct_0(A),B,C),k1_connsp_3(A,B)) ) ) ) ) ) ).
fof(t15_connsp_3,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)
& r2_hidden(C,B) )
=> k1_connsp_1(A,C) = k1_connsp_3(A,B) ) ) ) ) ).
fof(t16_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(C,A) )
=> ( r1_xboole_0(B,C)
| ( r1_tarski(k4_subset_1(u1_struct_0(A),B,C),k1_connsp_3(A,B))
& r1_tarski(k4_subset_1(u1_struct_0(A),B,C),k1_connsp_3(A,C))
& r1_tarski(B,k1_connsp_3(A,C))
& r1_tarski(C,k1_connsp_3(A,B)) ) ) ) ) ) ) ).
fof(t17_connsp_3,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)))
=> ( v2_connsp_1(B,A)
=> ( B = k1_xboole_0
| r1_tarski(k6_pre_topc(A,B),k1_connsp_3(A,B)) ) ) ) ) ).
fof(t18_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r3_connsp_1(A,B)
& v2_connsp_1(C,A)
& r1_xboole_0(B,C) )
=> ( C = k1_xboole_0
| r1_xboole_0(B,k1_connsp_3(A,C)) ) ) ) ) ) ).
fof(d2_connsp_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_connsp_3(B,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)
=> r3_connsp_1(A,D) ) )
& B = k5_setfam_1(u1_struct_0(A),C) ) ) ) ) ).
fof(t19_connsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> m1_connsp_3(k1_pre_topc(A),A) ) ).
fof(t20_connsp_3,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)))
=> ( B = u1_struct_0(A)
=> m1_connsp_3(B,A) ) ) ) ).
fof(t21_connsp_3,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))
=> ( ( r2_hidden(C,B)
& m1_connsp_3(B,A) )
=> r1_tarski(k1_connsp_1(A,C),B) ) ) ) ) ).
fof(t22_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( m1_connsp_3(B,A)
& m1_connsp_3(C,A) )
=> ( m1_connsp_3(k4_subset_1(u1_struct_0(A),B,C),A)
& m1_connsp_3(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ) ).
fof(t23_connsp_3,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)
=> m1_connsp_3(C,A) ) )
=> m1_connsp_3(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t24_connsp_3,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)
=> m1_connsp_3(C,A) ) )
=> m1_connsp_3(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t25_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( m1_connsp_3(B,A)
& m1_connsp_3(C,A) )
=> m1_connsp_3(k6_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(d3_connsp_3,axiom,
! [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))
=> ( r2_hidden(C,B)
=> k2_connsp_3(A,B,C) = C ) ) ) ) ).
fof(d4_connsp_3,axiom,
! [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(k3_pre_topc(A,B)))
=> ( B != k1_xboole_0
=> k3_connsp_3(A,B,C) = C ) ) ) ) ).
fof(d5_connsp_3,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)))
=> k4_connsp_3(A,B,C) = k5_subset_1(u1_struct_0(A),B,C) ) ) ) ).
fof(d6_connsp_3,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(k3_pre_topc(A,B))))
=> k5_connsp_3(A,B,C) = C ) ) ) ).
fof(d7_connsp_3,axiom,
! [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))
=> ( r2_hidden(C,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( D = k6_connsp_3(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k3_pre_topc(A,B)))
=> ( E = C
=> D = k1_connsp_1(k3_pre_topc(A,B),E) ) ) ) ) ) ) ) ) ).
fof(t26_connsp_3,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))
=> ( r2_hidden(C,B)
=> r2_hidden(C,k6_connsp_3(A,B,C)) ) ) ) ) ).
fof(t27_connsp_3,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))
=> ( r2_hidden(C,B)
=> k6_connsp_3(A,B,C) = k1_connsp_1(k3_pre_topc(A,B),k2_connsp_3(A,B,C)) ) ) ) ) ).
fof(t28_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(B,C)
=> k4_connsp_3(A,B,C) = B ) ) ) ) ).
fof(t29_connsp_3,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)))
=> ( v3_pre_topc(B,A)
=> v3_pre_topc(k4_connsp_3(A,B,C),k3_pre_topc(A,C)) ) ) ) ) ).
fof(t30_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(B,C)
=> k6_pre_topc(k3_pre_topc(A,C),k4_connsp_3(A,B,C)) = k5_subset_1(u1_struct_0(A),k6_pre_topc(A,B),C) ) ) ) ) ).
fof(t31_connsp_3,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,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
=> k6_pre_topc(k3_pre_topc(A,B),C) = k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k5_connsp_3(A,B,C)),B) ) ) ) ).
fof(t32_connsp_3,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(B,C)
=> r1_tarski(k6_pre_topc(k3_pre_topc(A,C),k4_connsp_3(A,B,C)),k6_pre_topc(A,B)) ) ) ) ) ).
fof(t33_connsp_3,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,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
=> ( r1_tarski(C,B)
=> k4_connsp_3(A,k5_connsp_3(A,B,C),B) = C ) ) ) ) ).
fof(t34_connsp_3,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(k3_pre_topc(A,C))))
=> ( ( B = D
& v2_connsp_1(D,k3_pre_topc(A,C)) )
=> v2_connsp_1(B,A) ) ) ) ) ) ).
fof(t35_connsp_3,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))
=> ( r2_hidden(C,B)
=> v2_connsp_1(k6_connsp_3(A,B,C),A) ) ) ) ) ).
fof(dt_m1_connsp_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_connsp_3(B,A)
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m1_connsp_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] : m1_connsp_3(B,A) ) ).
fof(dt_k1_connsp_3,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k1_connsp_3(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_connsp_3,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k2_connsp_3(A,B,C),u1_struct_0(k3_pre_topc(A,B))) ) ).
fof(dt_k3_connsp_3,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(k3_pre_topc(A,B))) )
=> m1_subset_1(k3_connsp_3(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k4_connsp_3,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))) )
=> m1_subset_1(k4_connsp_3(A,B,C),k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C)))) ) ).
fof(dt_k5_connsp_3,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(k3_pre_topc(A,B)))) )
=> m1_subset_1(k5_connsp_3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k6_connsp_3,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k6_connsp_3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
%------------------------------------------------------------------------------