SET007 Axioms: SET007+879.ax
%------------------------------------------------------------------------------
% File : SET007+879 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Boundary and Derivative of a Set
% 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 : topgen_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 101 ( 0 unt; 0 def)
% Number of atoms : 605 ( 55 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 586 ( 82 ~; 0 |; 238 &)
% ( 31 <=>; 235 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 38 ( 37 usr; 0 prp; 1-3 aty)
% Number of functors : 27 ( 27 usr; 6 con; 0-3 aty)
% Number of variables : 232 ( 218 !; 14 ?)
% 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_topgen_1,axiom,
! [A] :
( ( v6_group_1(A)
& l1_struct_0(A) )
=> ( v1_finset_1(k2_pre_topc(A))
& v1_card_4(k2_pre_topc(A)) ) ) ).
fof(cc1_topgen_1,axiom,
! [A] :
( l1_struct_0(A)
=> ( v6_group_1(A)
=> v1_orders_4(A) ) ) ).
fof(rc1_topgen_1,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A)
& v1_orders_4(A) ) ).
fof(rc2_topgen_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_orders_4(A) ) ).
fof(fc2_topgen_1,axiom,
! [A] :
( ( v1_orders_4(A)
& l1_struct_0(A) )
=> v1_card_4(k2_pre_topc(A)) ) ).
fof(rc3_topgen_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_urysohn1(A)
& v2_t_0topsp(A) ) ).
fof(fc3_topgen_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_urysohn1(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v4_pre_topc(k3_topgen_1(A,B),A) ) ).
fof(fc4_topgen_1,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v3_pre_topc(B,A)
& v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_membered(k2_tops_1(A,B))
& v2_membered(k2_tops_1(A,B))
& v3_membered(k2_tops_1(A,B))
& v4_membered(k2_tops_1(A,B))
& v5_membered(k2_tops_1(A,B))
& v1_xboole_0(k2_tops_1(A,B))
& v3_pre_topc(k2_tops_1(A,B),A)
& v4_pre_topc(k2_tops_1(A,B),A)
& v2_tops_1(k2_tops_1(A,B),A)
& v3_tops_1(k2_tops_1(A,B),A) ) ) ).
fof(rc4_topgen_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v3_pre_topc(B,A) ) ) ).
fof(rc5_topgen_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v4_pre_topc(B,A) ) ) ).
fof(fc5_topgen_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& l1_pre_topc(A)
& ~ v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k2_tops_1(A,B)) ) ).
fof(fc6_topgen_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v1_tdlat_3(A)
& l1_pre_topc(A)
& ~ v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k2_tops_1(A,B)) ) ).
fof(cc2_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v5_topgen_1(B,A)
=> ( v4_pre_topc(B,A)
& v2_topgen_1(B,A) ) ) ) ) ).
fof(cc3_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(B,A)
& v2_topgen_1(B,A) )
=> v5_topgen_1(B,A) ) ) ) ).
fof(cc4_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_xboole_0(B)
=> ( v4_pre_topc(B,A)
& v2_topgen_1(B,A)
& v5_topgen_1(B,A) ) ) ) ) ).
fof(rc6_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v4_pre_topc(B,A)
& v2_topgen_1(B,A)
& v5_topgen_1(B,A) ) ) ).
fof(cc5_topgen_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)))
=> ( ( ~ v1_xboole_0(B)
& v6_topgen_1(B,A) )
=> ~ v2_topgen_1(B,A) ) ) ) ).
fof(cc6_topgen_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)))
=> ( ( ~ v1_xboole_0(B)
& v2_topgen_1(B,A) )
=> ~ v6_topgen_1(B,A) ) ) ) ).
fof(cc7_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_xboole_0(B)
=> v6_topgen_1(B,A) ) ) ) ).
fof(fc7_topgen_1,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_membered(k2_tops_1(A,B))
& v2_membered(k2_tops_1(A,B))
& v3_membered(k2_tops_1(A,B))
& v4_membered(k2_tops_1(A,B))
& v5_membered(k2_tops_1(A,B))
& v1_xboole_0(k2_tops_1(A,B))
& v3_pre_topc(k2_tops_1(A,B),A)
& v4_pre_topc(k2_tops_1(A,B),A)
& v2_tops_1(k2_tops_1(A,B),A)
& v3_tops_1(k2_tops_1(A,B),A)
& v2_topgen_1(k2_tops_1(A,B),A)
& v5_topgen_1(k2_tops_1(A,B),A)
& v6_topgen_1(k2_tops_1(A,B),A) ) ) ).
fof(cc8_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(B,A)
& v4_pre_topc(B,A) ) ) ) ).
fof(fc8_topgen_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_membered(k3_topgen_1(A,B))
& v2_membered(k3_topgen_1(A,B))
& v3_membered(k3_topgen_1(A,B))
& v4_membered(k3_topgen_1(A,B))
& v5_membered(k3_topgen_1(A,B))
& v1_xboole_0(k3_topgen_1(A,B))
& v3_pre_topc(k3_topgen_1(A,B),A)
& v4_pre_topc(k3_topgen_1(A,B),A)
& v2_tops_1(k3_topgen_1(A,B),A)
& v3_tops_1(k3_topgen_1(A,B),A)
& v2_topgen_1(k3_topgen_1(A,B),A)
& v5_topgen_1(k3_topgen_1(A,B),A)
& v6_topgen_1(k3_topgen_1(A,B),A) ) ) ).
fof(cc9_topgen_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( v2_pre_topc(A)
& v1_orders_4(A) )
=> ( v2_pre_topc(A)
& v7_topgen_1(A) ) ) ) ).
fof(t1_topgen_1,axiom,
! [A] :
( l1_struct_0(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_xboole_0(B,k3_subset_1(u1_struct_0(A),C))
& k1_topgen_1(u1_struct_0(A),B,C) = k1_xboole_0 )
& ~ ( k1_topgen_1(u1_struct_0(A),B,C) != k1_xboole_0
& r1_xboole_0(B,k3_subset_1(u1_struct_0(A),C)) ) ) ) ) ) ).
fof(t2_topgen_1,axiom,
! [A] :
( l1_struct_0(A)
=> ( v1_orders_4(A)
<=> v1_card_4(k2_pre_topc(A)) ) ) ).
fof(t3_topgen_1,axiom,
! [A] :
( l1_struct_0(A)
=> ( v1_orders_4(A)
<=> r1_tarski(k1_card_1(k2_pre_topc(A)),k3_card_1(np__0)) ) ) ).
fof(t4_topgen_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_subset_1(u1_struct_0(A),B,k2_pre_topc(A)) = k2_pre_topc(A) ) ) ).
fof(t5_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_tops_1(A,B) = k3_subset_1(u1_struct_0(A),k6_pre_topc(A,k3_subset_1(u1_struct_0(A),B))) ) ) ).
fof(d1_topgen_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(k1_zfmisc_1(u1_struct_0(A))))
=> ( C = k2_topgen_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 = k2_tops_1(A,E)
& r2_hidden(E,B) ) ) ) ) ) ) ) ).
fof(t6_topgen_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
=> k2_topgen_1(A,B) = k1_xboole_0 ) ) ) ).
fof(t7_topgen_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)))
=> ( B = k1_tarski(C)
=> k2_topgen_1(A,B) = k1_tarski(k2_tops_1(A,C)) ) ) ) ) ).
fof(t8_topgen_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(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_tarski(B,C)
=> r1_tarski(k2_topgen_1(A,B),k2_topgen_1(A,C)) ) ) ) ) ).
fof(t9_topgen_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(k1_zfmisc_1(u1_struct_0(A))))
=> k2_topgen_1(A,k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C)) = k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),k2_topgen_1(A,B),k2_topgen_1(A,C)) ) ) ) ).
fof(t10_topgen_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k2_tops_1(A,B) = k1_topgen_1(u1_struct_0(A),k6_pre_topc(A,B),k1_tops_1(A,B)) ) ) ).
fof(t11_topgen_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))
=> ( r2_hidden(C,k2_tops_1(A,B))
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(D,A)
& r2_hidden(C,D) )
=> ( ~ r1_xboole_0(B,D)
& k1_topgen_1(u1_struct_0(A),D,B) != k1_xboole_0 ) ) ) ) ) ) ) ).
fof(t12_topgen_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))
=> ( r2_hidden(C,k2_tops_1(A,B))
<=> ! [D] :
( m1_yellow_8(D,A,C)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,D)
=> ( ~ r1_xboole_0(B,E)
& k1_topgen_1(u1_struct_0(A),E,B) != k1_xboole_0 ) ) ) ) ) ) ) ) ).
fof(t13_topgen_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))
=> ( r2_hidden(C,k2_tops_1(A,B))
<=> ? [D] :
( m1_yellow_8(D,A,C)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,D)
=> ( ~ r1_xboole_0(B,E)
& k1_topgen_1(u1_struct_0(A),E,B) != k1_xboole_0 ) ) ) ) ) ) ) ) ).
fof(t14_topgen_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)))
=> r1_tarski(k2_tops_1(A,k5_subset_1(u1_struct_0(A),B,C)),k4_subset_1(u1_struct_0(A),k5_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k2_tops_1(A,C)),k5_subset_1(u1_struct_0(A),k2_tops_1(A,B),k6_pre_topc(A,C)))) ) ) ) ).
fof(t15_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> u1_struct_0(A) = k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),k1_tops_1(A,B),k2_tops_1(A,B)),k1_tops_1(A,k3_subset_1(u1_struct_0(A),B))) ) ) ).
fof(t16_topgen_1,axiom,
! [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)
& v4_pre_topc(B,A) )
<=> k2_tops_1(A,B) = k1_xboole_0 ) ) ) ).
fof(d2_topgen_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( r1_topgen_1(A,B,C)
<=> r2_hidden(C,k6_pre_topc(A,k1_topgen_1(u1_struct_0(A),B,k1_tarski(C)))) ) ) ) ).
fof(t17_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( r1_topgen_1(A,B,C)
=> m1_subset_1(C,u1_struct_0(A)) ) ) ) ).
fof(d3_topgen_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)))
=> ( C = k3_topgen_1(A,B)
<=> ! [D] :
( r2_hidden(D,u1_struct_0(A))
=> ( r2_hidden(D,C)
<=> r1_topgen_1(A,B,D) ) ) ) ) ) ) ).
fof(t18_topgen_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] :
( r2_hidden(C,k3_topgen_1(A,B))
<=> r1_topgen_1(A,B,C) ) ) ) ).
fof(t19_topgen_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))
=> ( r2_hidden(C,k3_topgen_1(A,B))
<=> ! [D] :
( ( v3_pre_topc(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r2_hidden(C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,k5_subset_1(u1_struct_0(A),B,D))
& C != E ) ) ) ) ) ) ) ) ).
fof(t20_topgen_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))
=> ( r2_hidden(C,k3_topgen_1(A,B))
<=> ! [D] :
( m1_yellow_8(D,A,C)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,D)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( r2_hidden(F,k5_subset_1(u1_struct_0(A),B,E))
& C != F ) ) ) ) ) ) ) ) ) ).
fof(t21_topgen_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))
=> ( r2_hidden(C,k3_topgen_1(A,B))
<=> ? [D] :
( m1_yellow_8(D,A,C)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,D)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( r2_hidden(F,k5_subset_1(u1_struct_0(A),B,E))
& C != F ) ) ) ) ) ) ) ) ) ).
fof(d4_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( r2_topgen_1(A,B,C)
<=> ( r2_hidden(C,B)
& ~ r1_topgen_1(A,B,C) ) ) ) ) ).
fof(t22_topgen_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] :
( r2_hidden(C,k1_topgen_1(u1_struct_0(A),B,k3_topgen_1(A,B)))
<=> r2_topgen_1(A,B,C) ) ) ) ).
fof(t23_topgen_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))
=> ( r1_topgen_1(A,B,C)
<=> ! [D] :
( ( v3_pre_topc(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r2_hidden(C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( E != C
& r2_hidden(E,B)
& r2_hidden(E,D) ) ) ) ) ) ) ) ) ).
fof(t24_topgen_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))
=> ( r2_topgen_1(A,B,C)
<=> ? [D] :
( v3_pre_topc(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& k5_subset_1(u1_struct_0(A),D,B) = k1_struct_0(A,C) ) ) ) ) ) ).
fof(d5_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_topgen_1(B,A)
<=> r2_topgen_1(A,k2_pre_topc(A),B) ) ) ) ).
fof(t25_topgen_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_topgen_1(B,A)
<=> v3_pre_topc(k1_struct_0(A,B),A) ) ) ) ).
fof(d6_topgen_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(k1_zfmisc_1(u1_struct_0(A))))
=> ( C = k4_topgen_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 = k3_topgen_1(A,E)
& r2_hidden(E,B) ) ) ) ) ) ) ) ).
fof(t26_topgen_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))))
=> ( B = k1_xboole_0
=> k4_topgen_1(A,B) = k1_xboole_0 ) ) ) ).
fof(t27_topgen_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)
=> k4_topgen_1(A,C) = k1_tarski(k3_topgen_1(A,B)) ) ) ) ) ).
fof(t28_topgen_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(k4_topgen_1(A,B),k4_topgen_1(A,C)) ) ) ) ) ).
fof(t29_topgen_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))))
=> k4_topgen_1(A,k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C)) = k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),k4_topgen_1(A,B),k4_topgen_1(A,C)) ) ) ) ).
fof(t30_topgen_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)))
=> r1_tarski(k3_topgen_1(A,B),k6_pre_topc(A,B)) ) ) ).
fof(t31_topgen_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) = k4_subset_1(u1_struct_0(A),B,k3_topgen_1(A,B)) ) ) ).
fof(t32_topgen_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)))
=> ( r1_tarski(B,C)
=> r1_tarski(k3_topgen_1(A,B),k3_topgen_1(A,C)) ) ) ) ) ).
fof(t33_topgen_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)))
=> k3_topgen_1(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k3_topgen_1(A,B),k3_topgen_1(A,C)) ) ) ) ).
fof(t34_topgen_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)))
=> ( v1_urysohn1(A)
=> r1_tarski(k3_topgen_1(A,k3_topgen_1(A,B)),k3_topgen_1(A,B)) ) ) ) ).
fof(t35_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_urysohn1(A)
=> k6_pre_topc(A,k3_topgen_1(A,B)) = k3_topgen_1(A,B) ) ) ) ).
fof(t36_topgen_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),k4_topgen_1(A,B)),k3_topgen_1(A,k5_setfam_1(u1_struct_0(A),B))) ) ) ).
fof(t37_topgen_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)))
=> ! [D] :
( ( r1_tarski(B,C)
& r1_topgen_1(A,B,D) )
=> r1_topgen_1(A,C,D) ) ) ) ) ).
fof(d7_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_topgen_1(B,A)
<=> r1_tarski(B,k3_topgen_1(A,B)) ) ) ) ).
fof(d8_topgen_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_topgen_1(A)
<=> v2_topgen_1(k2_pre_topc(A),A) ) ) ).
fof(t38_topgen_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)))
=> ( ( v1_urysohn1(A)
& v2_topgen_1(B,A) )
=> v2_topgen_1(k6_pre_topc(A,B),A) ) ) ) ).
fof(d9_topgen_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))))
=> ( v4_topgen_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v2_topgen_1(C,A) ) ) ) ) ) ).
fof(t39_topgen_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))))
=> ( v4_topgen_1(B,A)
=> r1_tarski(k5_setfam_1(u1_struct_0(A),B),k5_setfam_1(u1_struct_0(A),k4_topgen_1(A,B))) ) ) ) ).
fof(t40_topgen_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))))
=> ( v4_topgen_1(B,A)
=> v2_topgen_1(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t41_topgen_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k2_tops_1(A,k1_pre_topc(A)) = k1_xboole_0 ) ).
fof(d10_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v5_topgen_1(B,A)
<=> ( v4_pre_topc(B,A)
& v2_topgen_1(B,A) ) ) ) ) ).
fof(t42_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> k3_topgen_1(A,k1_pre_topc(A)) = k1_pre_topc(A) ) ).
fof(t43_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v5_topgen_1(B,A)
<=> k3_topgen_1(A,B) = B ) ) ) ).
fof(t44_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v5_topgen_1(k1_pre_topc(A),A) ) ).
fof(d11_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v6_topgen_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ~ v1_xboole_0(C)
& r1_tarski(C,B)
& v2_topgen_1(C,A) ) ) ) ) ) ).
fof(t45_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v6_topgen_1(k1_pre_topc(A),A) ) ).
fof(t46_topgen_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ ( v1_urysohn1(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_subset_1(u1_struct_0(A),B,C) = k2_pre_topc(A)
& r1_xboole_0(B,C)
& v5_topgen_1(B,A)
& v6_topgen_1(C,A) ) ) ) ) ) ).
fof(t47_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_topgen_1(A,B) = k1_xboole_0 ) ) ).
fof(t48_topgen_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tops_1(B,A)
=> B = k2_pre_topc(A) ) ) ) ).
fof(d12_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( v1_card_1(B)
=> ( B = k5_topgen_1(A)
<=> ( ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& v1_tops_1(C,A)
& B = k1_card_1(C) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tops_1(C,A)
=> r1_tarski(B,k1_card_1(C)) ) ) ) ) ) ) ).
fof(d13_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v7_topgen_1(A)
<=> r1_tarski(k5_topgen_1(A),k3_card_1(np__0)) ) ) ).
fof(t49_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& v1_orders_4(A)
& l1_pre_topc(A) )
=> v7_topgen_1(A) ) ).
fof(t50_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k3_numbers
=> k3_subset_1(u1_struct_0(k3_topmetr),A) = k1_borsuk_5 ) ) ).
fof(t51_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k1_borsuk_5
=> k3_subset_1(u1_struct_0(k3_topmetr),A) = k3_numbers ) ) ).
fof(t52_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k3_numbers
=> k1_tops_1(k3_topmetr,A) = k1_xboole_0 ) ) ).
fof(t53_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k1_borsuk_5
=> k1_tops_1(k3_topmetr,A) = k1_xboole_0 ) ) ).
fof(t54_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k3_numbers
=> v1_tops_1(A,k3_topmetr) ) ) ).
fof(t55_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k1_borsuk_5
=> v1_tops_1(A,k3_topmetr) ) ) ).
fof(t56_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k3_numbers
=> v2_tops_1(A,k3_topmetr) ) ) ).
fof(t57_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k1_borsuk_5
=> v2_tops_1(A,k3_topmetr) ) ) ).
fof(t58_topgen_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ~ ( A = k1_numbers
& v2_tops_1(A,k3_topmetr) ) ) ).
fof(t59_topgen_1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
& ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
& v2_tops_1(A,k3_topmetr)
& v2_tops_1(B,k3_topmetr)
& ~ v2_tops_1(k4_subset_1(u1_struct_0(k3_topmetr),A,B),k3_topmetr) ) ) ).
fof(dt_k1_topgen_1,axiom,
! [A,B,C] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> m1_subset_1(k1_topgen_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(redefinition_k1_topgen_1,axiom,
! [A,B,C] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k1_topgen_1(A,B,C) = k4_xboole_0(B,C) ) ).
fof(dt_k2_topgen_1,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> m1_subset_1(k2_topgen_1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k3_topgen_1,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k3_topgen_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k4_topgen_1,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> m1_subset_1(k4_topgen_1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k5_topgen_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v1_card_1(k5_topgen_1(A)) ) ).
%------------------------------------------------------------------------------