SET007 Axioms: SET007+301.ax
%------------------------------------------------------------------------------
% File : SET007+301 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Paracompact and Metrizable 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 : pcomps_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 65 ( 11 unt; 0 def)
% Number of atoms : 365 ( 29 equ)
% Maximal formula atoms : 25 ( 5 avg)
% Number of connectives : 352 ( 52 ~; 0 |; 150 &)
% ( 11 <=>; 139 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 38 ( 36 usr; 1 prp; 0-3 aty)
% Number of functors : 31 ( 31 usr; 6 con; 0-5 aty)
% Number of variables : 144 ( 133 !; 11 ?)
% 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_pcomps_1,axiom,
! [A] :
( v1_pre_topc(k2_pcomps_1(A))
& v2_pre_topc(k2_pcomps_1(A)) ) ).
fof(fc2_pcomps_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k2_pcomps_1(A))
& v1_pre_topc(k2_pcomps_1(A))
& v2_pre_topc(k2_pcomps_1(A)) ) ) ).
fof(rc1_pcomps_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_pcomps_1(A) ) ).
fof(fc3_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ( v1_pre_topc(k5_pcomps_1(A))
& v2_pre_topc(k5_pcomps_1(A)) ) ) ).
fof(fc4_pcomps_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( ~ v3_struct_0(k5_pcomps_1(A))
& v1_pre_topc(k5_pcomps_1(A))
& v2_pre_topc(k5_pcomps_1(A)) ) ) ).
fof(rc2_pcomps_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& v3_compts_1(A) ) ).
fof(rc3_pcomps_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& v3_pcomps_1(A) ) ).
fof(t1_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_xreal_0(C,D)
=> r1_tarski(k9_metric_1(A,B,C),k9_metric_1(A,B,D)) ) ) ) ) ) ).
fof(t2_pcomps_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ ( k6_pre_topc(A,B) != k1_xboole_0
& B = k1_xboole_0 )
& ~ ( B != k1_xboole_0
& k6_pre_topc(A,B) = k1_xboole_0 ) ) ) ) ).
fof(t3_pcomps_1,axiom,
$true ).
fof(t4_pcomps_1,axiom,
$true ).
fof(t5_pcomps_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))))
=> ( r1_pre_topc(A,B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(C,D)
& r2_hidden(D,B) ) ) ) ) ) ).
fof(d1_pcomps_1,axiom,
! [A] : k2_pcomps_1(A) = g1_pre_topc(A,k1_pcomps_1(A)) ).
fof(t6_pcomps_1,axiom,
$true ).
fof(t7_pcomps_1,axiom,
! [A] : u1_pre_topc(k2_pcomps_1(A)) = k1_pcomps_1(A) ).
fof(t8_pcomps_1,axiom,
! [A] : u1_struct_0(k2_pcomps_1(A)) = A ).
fof(t9_pcomps_1,axiom,
! [A] : v2_compts_1(k2_pcomps_1(k1_tarski(A))) ).
fof(t10_pcomps_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v3_compts_1(A)
=> v4_pre_topc(k1_struct_0(A,B),A) ) ) ) ).
fof(t12_pcomps_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(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( r1_tarski(B,C)
& v1_pcomps_1(C,A) )
=> v1_pcomps_1(B,A) ) ) ) ) ).
fof(t13_pcomps_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))))
=> ( v1_finset_1(B)
=> v1_pcomps_1(B,A) ) ) ) ).
fof(d3_pcomps_1,axiom,
! [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(k1_zfmisc_1(u1_struct_0(A))))
=> ( C = k3_pcomps_1(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(D,C)
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& D = k6_pre_topc(A,E)
& r2_hidden(E,B) ) ) ) ) ) ) ) ).
fof(t14_pcomps_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))))
=> v2_tops_2(k3_pcomps_1(A,B),A) ) ) ).
fof(t15_pcomps_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))))
=> ( B = k1_xboole_0
=> k3_pcomps_1(A,B) = k1_xboole_0 ) ) ) ).
fof(t16_pcomps_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,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( C = k1_tarski(B)
=> k3_pcomps_1(A,C) = k1_tarski(k6_pre_topc(A,B)) ) ) ) ) ).
fof(t17_pcomps_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(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_tarski(B,C)
=> r1_tarski(k3_pcomps_1(A,B),k3_pcomps_1(A,C)) ) ) ) ) ).
fof(t18_pcomps_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(k1_zfmisc_1(u1_struct_0(A))))
=> k3_pcomps_1(A,k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C)) = k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),k3_pcomps_1(A,B),k3_pcomps_1(A,C)) ) ) ) ).
fof(t19_pcomps_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))))
=> ( v1_finset_1(B)
=> k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)) ) ) ) ).
fof(t20_pcomps_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))))
=> r1_setfam_1(B,k3_pcomps_1(A,B)) ) ) ).
fof(t21_pcomps_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))))
=> ( v1_pcomps_1(B,A)
=> v1_pcomps_1(k3_pcomps_1(A,B),A) ) ) ) ).
fof(t22_pcomps_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))))
=> r1_tarski(k5_setfam_1(u1_struct_0(A),B),k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B))) ) ) ).
fof(t23_pcomps_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))))
=> ( v1_pcomps_1(B,A)
=> k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)) ) ) ) ).
fof(t24_pcomps_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))))
=> ( ( v1_pcomps_1(B,A)
& v2_tops_2(B,A) )
=> v4_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(d4_pcomps_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v2_pcomps_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_pre_topc(A,B)
& v1_tops_2(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_tops_2(C,A)
& r1_pre_topc(A,C)
& r1_setfam_1(C,B)
& v1_pcomps_1(C,A) ) ) ) ) ) ) ).
fof(t25_pcomps_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_compts_1(A)
=> v2_pcomps_1(A) ) ) ).
fof(t26_pcomps_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,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v2_pcomps_1(A)
& v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& r1_xboole_0(B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,C)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(E,A)
& v3_pre_topc(F,A)
& r1_tarski(B,E)
& r2_hidden(D,F)
& r1_xboole_0(E,F) ) ) ) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r1_tarski(B,D)
& r1_tarski(C,E)
& r1_xboole_0(D,E) ) ) ) ) ) ) ) ).
fof(t27_pcomps_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v3_compts_1(A)
& v2_pcomps_1(A) )
=> v4_compts_1(A) ) ) ).
fof(t28_pcomps_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v3_compts_1(A)
& v2_pcomps_1(A) )
=> v5_compts_1(A) ) ) ).
fof(d5_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( B = k4_pcomps_1(A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,C)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& r1_tarski(k9_metric_1(A,D,E),C) ) ) ) ) ) ) ) ) ) ).
fof(t29_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& r1_tarski(k9_metric_1(A,B,C),u1_struct_0(A)) ) ) ) ).
fof(t30_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( v9_metric_1(A)
& r2_hidden(B,k9_metric_1(A,C,D))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& r1_tarski(k9_metric_1(A,B,E),k9_metric_1(A,C,D)) ) ) ) ) ) ) ) ).
fof(t31_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( v9_metric_1(A)
& r2_hidden(D,k5_subset_1(u1_struct_0(A),k9_metric_1(A,E,B),k9_metric_1(A,F,C)))
& ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( r1_tarski(k9_metric_1(A,D,G),k9_metric_1(A,E,B))
& r1_tarski(k9_metric_1(A,D,G),k9_metric_1(A,F,C)) ) ) ) ) ) ) ) ) ) ).
fof(t32_pcomps_1,axiom,
$true ).
fof(t33_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_xreal_0(C)
=> ( v9_metric_1(A)
=> r2_hidden(k9_metric_1(A,B,C),k4_pcomps_1(A)) ) ) ) ) ).
fof(t34_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> r2_hidden(u1_struct_0(A),k4_pcomps_1(A)) ) ).
fof(t35_pcomps_1,axiom,
! [A] :
( l1_metric_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)))
=> ( ( r2_hidden(B,k4_pcomps_1(A))
& r2_hidden(C,k4_pcomps_1(A)) )
=> r2_hidden(k5_subset_1(u1_struct_0(A),B,C),k4_pcomps_1(A)) ) ) ) ) ).
fof(t36_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_tarski(B,k4_pcomps_1(A))
=> r2_hidden(k5_setfam_1(u1_struct_0(A),B),k4_pcomps_1(A)) ) ) ) ).
fof(t37_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> ( v2_pre_topc(g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A)))
& l1_pre_topc(g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A))) ) ) ).
fof(d6_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> k5_pcomps_1(A) = g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A)) ) ).
fof(t38_pcomps_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> v3_compts_1(k5_pcomps_1(A)) ) ).
fof(d7_pcomps_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( r1_pcomps_1(A,B)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( ( k1_metric_1(A,A,B,C,D) = np__0
=> C = D )
& ( C = D
=> k1_metric_1(A,A,B,C,D) = np__0 )
& k1_metric_1(A,A,B,C,D) = k1_metric_1(A,A,B,D,C)
& r1_xreal_0(k1_metric_1(A,A,B,C,E),k3_real_1(k1_metric_1(A,A,B,C,D),k1_metric_1(A,A,B,D,E))) ) ) ) ) ) ) ).
fof(t39_pcomps_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( r1_pcomps_1(A,B)
<=> ( v6_metric_1(g1_metric_1(A,B))
& v7_metric_1(g1_metric_1(A,B))
& v8_metric_1(g1_metric_1(A,B))
& v9_metric_1(g1_metric_1(A,B))
& l1_metric_1(g1_metric_1(A,B)) ) ) ) ).
fof(d8_pcomps_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( r1_pcomps_1(A,B)
=> k6_pcomps_1(A,B) = g1_metric_1(A,B) ) ) ) ).
fof(d9_pcomps_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v3_pcomps_1(A)
<=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& r1_pcomps_1(u1_struct_0(A),B)
& k4_pcomps_1(k6_pcomps_1(u1_struct_0(A),B)) = u1_pre_topc(A) ) ) ) ).
fof(s1_pcomps_1,axiom,
( ! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(f1_s1_pcomps_1)))
=> ( r2_hidden(A,f2_s1_pcomps_1)
=> r2_hidden(f4_s1_pcomps_1(A),f3_s1_pcomps_1) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f2_s1_pcomps_1,f3_s1_pcomps_1)
& m2_relset_1(A,f2_s1_pcomps_1,f3_s1_pcomps_1)
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(f1_s1_pcomps_1)))
=> ( r2_hidden(B,f2_s1_pcomps_1)
=> k1_funct_1(A,B) = f4_s1_pcomps_1(B) ) ) ) ) ).
fof(dt_k1_pcomps_1,axiom,
! [A] : m1_subset_1(k1_pcomps_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) ).
fof(redefinition_k1_pcomps_1,axiom,
! [A] : k1_pcomps_1(A) = k1_zfmisc_1(A) ).
fof(dt_k2_pcomps_1,axiom,
! [A] : l1_pre_topc(k2_pcomps_1(A)) ).
fof(dt_k3_pcomps_1,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> m1_subset_1(k3_pcomps_1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k4_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> m1_subset_1(k4_pcomps_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k5_pcomps_1,axiom,
! [A] :
( l1_metric_1(A)
=> l1_pre_topc(k5_pcomps_1(A)) ) ).
fof(dt_k6_pcomps_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( ~ v3_struct_0(k6_pcomps_1(A,B))
& v1_metric_1(k6_pcomps_1(A,B))
& v6_metric_1(k6_pcomps_1(A,B))
& v7_metric_1(k6_pcomps_1(A,B))
& v8_metric_1(k6_pcomps_1(A,B))
& v9_metric_1(k6_pcomps_1(A,B))
& l1_metric_1(k6_pcomps_1(A,B)) ) ) ).
fof(d2_pcomps_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_pcomps_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(C,D)
& v3_pre_topc(D,A)
& v1_finset_1(a_3_0_pcomps_1(A,B,D)) ) ) ) ) ) ).
fof(t11_pcomps_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)))
=> r1_tarski(a_3_1_pcomps_1(A,B,C),B) ) ) ) ).
fof(fraenkel_a_3_0_pcomps_1,axiom,
! [A,B,C,D] :
( ( l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_0_pcomps_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& A = E
& r2_hidden(E,C)
& ~ r1_xboole_0(E,D) ) ) ) ).
fof(fraenkel_a_3_1_pcomps_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_1_pcomps_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& A = E
& r2_hidden(E,C)
& ~ r1_xboole_0(E,D) ) ) ) ).
%------------------------------------------------------------------------------