SET007 Axioms: SET007+915.ax
%------------------------------------------------------------------------------
% File : SET007+915 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Borel Families of Subsets of Topological 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 : topgen_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 124 ( 4 unt; 0 def)
% Number of atoms : 768 ( 39 equ)
% Maximal formula atoms : 25 ( 6 avg)
% Number of connectives : 776 ( 132 ~; 1 |; 373 &)
% ( 23 <=>; 247 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 63 ( 62 usr; 0 prp; 1-3 aty)
% Number of functors : 45 ( 45 usr; 7 con; 0-4 aty)
% Number of variables : 257 ( 245 !; 12 ?)
% 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_4,axiom,
( v1_membered(k3_numbers)
& v2_membered(k3_numbers)
& v3_membered(k3_numbers)
& ~ v1_xboole_0(k3_numbers)
& ~ v1_finset_1(k3_numbers)
& v1_card_4(k3_numbers) ) ).
fof(cc1_topgen_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v6_group_1(A)
=> v5_waybel23(A) ) ) ).
fof(fc2_topgen_4,axiom,
( v1_membered(k1_numbers)
& v2_membered(k1_numbers)
& ~ v1_xboole_0(k1_numbers)
& ~ v1_finset_1(k1_numbers)
& ~ v1_card_4(k1_numbers) ) ).
fof(cc2_topgen_4,axiom,
! [A] :
( ~ v1_card_4(A)
=> ~ v1_finset_1(A) ) ).
fof(cc3_topgen_4,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ~ v1_realset1(A) ) ).
fof(rc1_topgen_4,axiom,
? [A] :
( ~ v1_xboole_0(A)
& ~ v1_card_4(A) ) ).
fof(cc4_topgen_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_waybel23(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v7_topgen_1(A) ) ) ) ).
fof(cc5_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A) )
=> ( ~ v1_xboole_0(B)
& v1_finsub_1(B)
& v2_finsub_1(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& v3_topgen_4(B,A) ) ) ) ).
fof(cc6_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v3_topgen_4(B,A)
=> ~ v1_xboole_0(B) ) ) ).
fof(cc7_topgen_4,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_prob_1(B,u1_struct_0(A))
& v1_topgen_4(B,A)
& v3_topgen_4(B,u1_struct_0(A)) )
=> ( v2_topgen_4(B,A)
& v4_topgen_4(B,u1_struct_0(A)) ) ) ) ) ).
fof(cc8_topgen_4,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_prob_1(B,u1_struct_0(A))
& v2_topgen_4(B,A)
& v4_topgen_4(B,u1_struct_0(A)) )
=> ( ~ v1_xboole_0(B)
& v1_topgen_4(B,A)
& v3_topgen_4(B,u1_struct_0(A)) ) ) ) ) ).
fof(fc3_topgen_4,axiom,
! [A,B] :
( ( v1_card_4(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> v1_card_4(k7_setfam_1(A,B)) ) ).
fof(cc9_topgen_4,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_xboole_0(B)
=> ( v1_tops_2(B,A)
& v2_tops_2(B,A) ) ) ) ) ).
fof(rc2_topgen_4,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_tops_2(B,A)
& v2_tops_2(B,A)
& v1_card_4(B) ) ) ).
fof(cc10_topgen_4,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_card_4(A) ) ).
fof(fc4_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_membered(k1_pre_topc(A))
& v2_membered(k1_pre_topc(A))
& v3_membered(k1_pre_topc(A))
& v4_membered(k1_pre_topc(A))
& v5_membered(k1_pre_topc(A))
& v1_xboole_0(k1_pre_topc(A))
& v3_pre_topc(k1_pre_topc(A),A)
& v4_pre_topc(k1_pre_topc(A),A)
& v2_tops_1(k1_pre_topc(A),A)
& v3_tops_1(k1_pre_topc(A),A)
& v2_connsp_1(k1_pre_topc(A),A)
& v1_ordinal1(k1_pre_topc(A))
& v2_ordinal1(k1_pre_topc(A))
& v3_ordinal1(k1_pre_topc(A))
& v1_finset_1(k1_pre_topc(A))
& v6_compts_1(k1_pre_topc(A),A)
& v1_card_4(k1_pre_topc(A))
& v2_topgen_1(k1_pre_topc(A),A)
& v5_topgen_1(k1_pre_topc(A),A)
& v6_topgen_1(k1_pre_topc(A),A)
& v5_topgen_4(k1_pre_topc(A),A)
& v6_topgen_4(k1_pre_topc(A),A) ) ) ).
fof(fc5_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_pre_topc(k2_pre_topc(A),A)
& v4_pre_topc(k2_pre_topc(A),A)
& v1_tops_1(k2_pre_topc(A),A)
& v5_topgen_4(k2_pre_topc(A),A)
& v6_topgen_4(k2_pre_topc(A),A) ) ) ).
fof(cc11_topgen_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( v2_pre_topc(A)
& v7_topgen_4(A) )
=> ( v2_pre_topc(A)
& v2_t_0topsp(A) ) ) ) ).
fof(cc12_topgen_4,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( v2_pre_topc(A)
& v1_urysohn1(A) )
=> ( v2_pre_topc(A)
& v7_topgen_4(A) ) ) ) ).
fof(fc6_topgen_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_card_4(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_membered(k4_topgen_4(A,B))
& v2_membered(k4_topgen_4(A,B))
& v3_membered(k4_topgen_4(A,B))
& v4_membered(k4_topgen_4(A,B))
& v5_membered(k4_topgen_4(A,B))
& v1_xboole_0(k4_topgen_4(A,B))
& v3_pre_topc(k4_topgen_4(A,B),A)
& v4_pre_topc(k4_topgen_4(A,B),A)
& v2_tops_1(k4_topgen_4(A,B),A)
& v3_tops_1(k4_topgen_4(A,B),A)
& v2_connsp_1(k4_topgen_4(A,B),A)
& v1_ordinal1(k4_topgen_4(A,B))
& v2_ordinal1(k4_topgen_4(A,B))
& v3_ordinal1(k4_topgen_4(A,B))
& v1_finset_1(k4_topgen_4(A,B))
& v6_compts_1(k4_topgen_4(A,B),A)
& v1_card_4(k4_topgen_4(A,B))
& v2_topgen_1(k4_topgen_4(A,B),A)
& v5_topgen_1(k4_topgen_4(A,B),A)
& v6_topgen_1(k4_topgen_4(A,B),A) ) ) ).
fof(rc3_topgen_4,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_waybel23(A)
& v7_topgen_1(A) ) ).
fof(fc7_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v1_xboole_0(k1_topgen_4(A))
& v1_prob_1(k1_topgen_4(A),u1_struct_0(A))
& v1_topgen_4(k1_topgen_4(A),A)
& v2_topgen_4(k1_topgen_4(A),A)
& v3_topgen_4(k1_topgen_4(A),u1_struct_0(A))
& v4_topgen_4(k1_topgen_4(A),u1_struct_0(A)) ) ) ).
fof(rc4_topgen_4,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_xboole_0(B)
& v1_prob_1(B,u1_struct_0(A))
& v1_topgen_4(B,A)
& v2_topgen_4(B,A)
& v3_topgen_4(B,u1_struct_0(A))
& v4_topgen_4(B,u1_struct_0(A)) ) ) ).
fof(fc8_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v1_xboole_0(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)))
& v1_finsub_1(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)))
& v2_finsub_1(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)))
& v1_prob_1(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)),u1_struct_0(A))
& v1_topgen_4(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)),A)
& v2_topgen_4(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)),A)
& v3_topgen_4(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)),u1_struct_0(A))
& v4_topgen_4(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)),u1_struct_0(A)) ) ) ).
fof(rc5_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& ~ v1_xboole_0(B)
& v1_finsub_1(B)
& v2_finsub_1(B)
& v1_prob_1(B,u1_struct_0(A))
& v1_measure1(B,u1_struct_0(A))
& v3_topgen_4(B,u1_struct_0(A)) ) ) ).
fof(cc13_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_prob_1(B,u1_struct_0(A))
=> ( ~ v1_xboole_0(B)
& v1_finsub_1(B)
& v2_finsub_1(B)
& v1_prob_1(B,u1_struct_0(A))
& v3_topgen_4(B,u1_struct_0(A)) ) ) ) ).
fof(rc6_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_prob_1(B,u1_struct_0(A))
& ~ v1_xboole_0(B)
& v1_finsub_1(B)
& v2_finsub_1(B)
& v1_prob_1(B,u1_struct_0(A))
& v1_topgen_4(B,A)
& v2_topgen_4(B,A)
& v3_topgen_4(B,u1_struct_0(A))
& v4_topgen_4(B,u1_struct_0(A)) ) ) ).
fof(fc9_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_tops_2(k1_openlatt(A),A)
& v1_topgen_4(k1_openlatt(A),A) ) ) ).
fof(cc14_topgen_4,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)))
=> ( v5_topgen_4(B,A)
=> v8_topgen_4(B,A) ) ) ) ).
fof(cc15_topgen_4,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)))
=> ( v6_topgen_4(B,A)
=> v8_topgen_4(B,A) ) ) ) ).
fof(d1_topgen_4,axiom,
! [A] :
( l1_struct_0(A)
=> k1_topgen_4(A) = k1_pcomps_1(u1_struct_0(A)) ) ).
fof(t1_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v1_card_4(B)
<=> v1_card_4(k7_setfam_1(A,B)) ) ) ).
fof(t3_topgen_4,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)))
=> ( v4_pre_topc(B,A)
<=> r1_tarski(k3_topgen_1(A,B),B) ) ) ) ).
fof(t4_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_cantor_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(C,A)
& C != k1_xboole_0
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(D,B)
& r1_tarski(D,C)
& D != k1_xboole_0 ) ) ) ) ) ) ).
fof(t5_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> r1_tarski(k5_topgen_1(A),k2_waybel23(A)) ) ).
fof(t6_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v7_topgen_1(A)
<=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_tops_1(B,A)
& v1_card_4(B) ) ) ) ).
fof(t7_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v5_waybel23(A)
=> v7_topgen_1(A) ) ) ).
fof(t8_topgen_4,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_connsp_1(A,B,C)
=> k2_tops_1(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k2_tops_1(A,B),k2_tops_1(A,C)) ) ) ) ) ).
fof(t9_topgen_4,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)
=> r1_tarski(k2_tops_1(A,k5_setfam_1(u1_struct_0(A),B)),k5_setfam_1(u1_struct_0(A),k2_topgen_1(A,B))) ) ) ) ).
fof(t10_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ( v7_topgen_1(A)
<=> r1_tarski(k1_card_1(k2_pre_topc(A)),k3_card_1(np__0)) ) ) ).
fof(t11_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ( v7_topgen_1(A)
<=> v1_orders_4(A) ) ) ).
fof(d2_topgen_4,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_topgen_4(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(C,A)
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d3_topgen_4,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))))
=> ( v2_topgen_4(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(C,A)
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d4_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v3_topgen_4(B,A)
<=> ! [C] :
( ( v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( r1_tarski(C,B)
=> r2_hidden(k5_setfam_1(A,C),B) ) ) ) ) ).
fof(t12_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v3_topgen_4(B,A)
=> r2_hidden(k1_xboole_0,B) ) ) ).
fof(t13_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
<=> ( v1_prob_1(B,A)
& v3_topgen_4(B,A) ) ) ) ).
fof(d5_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v4_topgen_4(B,A)
<=> ! [C] :
( ( v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( r1_tarski(C,B)
=> r2_hidden(k6_setfam_1(A,C),B) ) ) ) ) ).
fof(t14_topgen_4,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))))
=> ( ( ( v2_topgen_4(B,A)
& v1_prob_1(B,u1_struct_0(A)) )
=> ( v1_topgen_4(B,A)
& v1_prob_1(B,u1_struct_0(A)) ) )
& ( ( v1_topgen_4(B,A)
& v1_prob_1(B,u1_struct_0(A)) )
=> ( v2_topgen_4(B,A)
& v1_prob_1(B,u1_struct_0(A)) ) ) ) ) ) ).
fof(t15_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v1_prob_1(B,A)
=> B = k7_setfam_1(A,B) ) ) ).
fof(t16_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( r1_tarski(B,C)
& v1_prob_1(C,A) )
=> r1_tarski(k7_setfam_1(A,B),C) ) ) ) ).
fof(t17_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( ( v4_topgen_4(B,A)
& v1_prob_1(B,A) )
=> ( v3_topgen_4(B,A)
& v1_prob_1(B,A) ) )
& ( ( v3_topgen_4(B,A)
& v1_prob_1(B,A) )
=> ( v4_topgen_4(B,A)
& v1_prob_1(B,A) ) ) ) ) ).
fof(t18_topgen_4,axiom,
! [A] :
( v1_xboole_0(k1_xboole_0)
& m1_subset_1(k1_xboole_0,k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(t19_topgen_4,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)
=> ( v3_pre_topc(B,A)
<=> v1_tops_2(C,A) ) ) ) ) ) ).
fof(t20_topgen_4,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)
=> ( v4_pre_topc(B,A)
<=> v2_tops_2(C,A) ) ) ) ) ) ).
fof(t21_topgen_4,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))))
=> ( ( v2_tops_2(B,A)
& v2_tops_2(C,A) )
=> v2_tops_2(k2_topgen_4(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t22_topgen_4,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))))
=> ( ( v2_tops_2(B,A)
& v2_tops_2(C,A) )
=> v2_tops_2(k3_topgen_4(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t23_topgen_4,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))))
=> ( ( v1_tops_2(B,A)
& v1_tops_2(C,A) )
=> v1_tops_2(k2_topgen_4(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t24_topgen_4,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))))
=> ( ( v1_tops_2(B,A)
& v1_tops_2(C,A) )
=> v1_tops_2(k3_topgen_4(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t25_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> r1_tarski(k1_card_1(k2_topgen_4(A,B,C)),k1_card_1(k12_mcart_1(k1_zfmisc_1(A),k1_zfmisc_1(A),B,C))) ) ) ).
fof(t26_topgen_4,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> r1_tarski(k1_card_1(k3_topgen_4(A,B,C)),k1_card_1(k12_mcart_1(k1_zfmisc_1(A),k1_zfmisc_1(A),B,C))) ) ) ).
fof(t27_topgen_4,axiom,
! [A,B] : r1_tarski(k3_tarski(k2_setfam_1(A,B)),k2_xboole_0(k3_tarski(A),k3_tarski(B))) ).
fof(t28_topgen_4,axiom,
! [A,B] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& k2_xboole_0(k3_tarski(A),k3_tarski(B)) != k3_tarski(k2_setfam_1(A,B)) ) ).
fof(t29_topgen_4,axiom,
! [A] : k2_setfam_1(k1_xboole_0,A) = k1_xboole_0 ).
fof(t30_topgen_4,axiom,
! [A,B] :
~ ( k2_setfam_1(A,B) = k1_xboole_0
& A != k1_xboole_0
& B != k1_xboole_0 ) ).
fof(t31_topgen_4,axiom,
! [A,B] :
~ ( k3_setfam_1(A,B) = k1_xboole_0
& A != k1_xboole_0
& B != k1_xboole_0 ) ).
fof(t32_topgen_4,axiom,
! [A,B] : r1_tarski(k1_setfam_1(k2_setfam_1(A,B)),k2_xboole_0(k1_setfam_1(A),k1_setfam_1(B))) ).
fof(t33_topgen_4,axiom,
! [A,B] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& k1_setfam_1(k2_setfam_1(A,B)) != k2_xboole_0(k1_setfam_1(A),k1_setfam_1(B)) ) ).
fof(t34_topgen_4,axiom,
! [A,B] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& k3_xboole_0(k1_setfam_1(A),k1_setfam_1(B)) != k1_setfam_1(k3_setfam_1(A,B)) ) ).
fof(d6_topgen_4,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)))
=> ( v5_topgen_4(B,A)
<=> ? [C] :
( v2_tops_2(C,A)
& v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& B = k5_setfam_1(u1_struct_0(A),C) ) ) ) ) ).
fof(d7_topgen_4,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)))
=> ( v6_topgen_4(B,A)
<=> ? [C] :
( v1_tops_2(C,A)
& v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& B = k6_setfam_1(u1_struct_0(A),C) ) ) ) ) ).
fof(t35_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v5_topgen_4(k1_pre_topc(A),A) ) ).
fof(t36_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v6_topgen_4(k1_pre_topc(A),A) ) ).
fof(t37_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v5_topgen_4(k2_pre_topc(A),A) ) ).
fof(t38_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> v6_topgen_4(k2_pre_topc(A),A) ) ).
fof(t39_topgen_4,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)))
=> ( v5_topgen_4(B,A)
=> v6_topgen_4(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t40_topgen_4,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)))
=> ( v6_topgen_4(B,A)
=> v5_topgen_4(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t41_topgen_4,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)))
=> ( ( v5_topgen_4(B,A)
& v5_topgen_4(C,A) )
=> v5_topgen_4(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t42_topgen_4,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)))
=> ( ( v5_topgen_4(B,A)
& v5_topgen_4(C,A) )
=> v5_topgen_4(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t43_topgen_4,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)))
=> ( ( v6_topgen_4(B,A)
& v6_topgen_4(C,A) )
=> v6_topgen_4(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t44_topgen_4,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)))
=> ( ( v6_topgen_4(B,A)
& v6_topgen_4(C,A) )
=> v6_topgen_4(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t45_topgen_4,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)))
=> ( v4_pre_topc(B,A)
=> v5_topgen_4(B,A) ) ) ) ).
fof(t46_topgen_4,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)))
=> ( v3_pre_topc(B,A)
=> v6_topgen_4(B,A) ) ) ) ).
fof(t47_topgen_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = k3_numbers
=> v5_topgen_4(A,k3_topmetr) ) ) ).
fof(d8_topgen_4,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v7_topgen_4(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> v4_pre_topc(k3_topgen_1(A,B),A) ) ) ) ).
fof(t48_topgen_4,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_urysohn1(A)
=> v7_topgen_4(A) ) ) ).
fof(t49_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v7_topgen_4(A)
=> v2_t_0topsp(A) ) ) ).
fof(t50_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r2_topgen_1(A,k2_pre_topc(A),B)
| r1_topgen_1(A,k2_pre_topc(A),B) ) ) ) ).
fof(d9_topgen_4,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_4(A,B,C)
<=> ! [D] :
( m1_connsp_2(D,A,C)
=> ~ v1_card_4(k5_subset_1(u1_struct_0(A),D,B)) ) ) ) ) ) ).
fof(t51_topgen_4,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] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r1_topgen_4(A,B,D)
& r1_tarski(B,C) )
=> r1_topgen_4(A,C,D) ) ) ) ) ) ).
fof(d10_topgen_4,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)))
=> ( C = k4_topgen_4(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,C)
<=> r1_topgen_4(A,B,D) ) ) ) ) ) ) ).
fof(t52_topgen_4,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_4(A,B,C)
=> r1_topgen_1(A,B,C) ) ) ) ) ).
fof(t53_topgen_4,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(k4_topgen_4(A,B),k3_topgen_1(A,B)) ) ) ).
fof(t54_topgen_4,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)))
=> k4_topgen_4(A,B) = k6_pre_topc(A,k4_topgen_4(A,B)) ) ) ).
fof(t55_topgen_4,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(k4_topgen_4(A,B),k4_topgen_4(A,C)) ) ) ) ) ).
fof(t56_topgen_4,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] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r1_topgen_4(A,k4_subset_1(u1_struct_0(A),B,C),D)
& ~ r1_topgen_4(A,B,D)
& ~ r1_topgen_4(A,C,D) ) ) ) ) ) ).
fof(t57_topgen_4,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)))
=> k4_topgen_4(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k4_topgen_4(A,B),k4_topgen_4(A,C)) ) ) ) ).
fof(t58_topgen_4,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_card_4(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ r1_topgen_4(A,B,C) ) ) ) ) ).
fof(t59_topgen_4,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_card_4(B)
=> k4_topgen_4(A,B) = k1_xboole_0 ) ) ) ).
fof(t60_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ ( v5_waybel23(A)
& ! [B] :
( m1_cantor_1(B,A)
=> ~ v1_card_4(B) ) ) ) ).
fof(t61_topgen_4,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( ~ v1_xboole_0(k1_topgen_3(k5_numbers,A,B))
& v1_card_4(k1_topgen_3(k5_numbers,A,B))
& m1_subset_1(k1_topgen_3(k5_numbers,A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(t62_topgen_4,axiom,
! [A,B] :
( m1_prob_1(B,A)
<=> ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(t63_topgen_4,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))))
=> ( ( v1_topgen_4(B,A)
& r1_tarski(B,C) )
=> v1_topgen_4(C,A) ) ) ) ) ).
fof(t64_topgen_4,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))))
=> ( ( v2_topgen_4(B,A)
& r1_tarski(B,C) )
=> v2_topgen_4(C,A) ) ) ) ) ).
fof(t65_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_topgen_4(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)),A)
& v1_prob_1(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)),u1_struct_0(A))
& v3_topgen_4(k11_prob_1(u1_struct_0(A),k1_topgen_4(A)),u1_struct_0(A)) ) ) ).
fof(t66_topgen_4,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_prob_1(B,u1_struct_0(A))
& v3_topgen_4(B,u1_struct_0(A)) )
=> m1_prob_1(B,u1_struct_0(A)) ) ) ) ).
fof(t67_topgen_4,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] :
( v1_prob_1(C,u1_struct_0(A))
& v1_topgen_4(C,A)
& v3_topgen_4(C,u1_struct_0(A))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& r1_tarski(B,C)
& ! [D] :
( ( v1_prob_1(D,u1_struct_0(A))
& v1_topgen_4(D,A)
& v3_topgen_4(D,u1_struct_0(A))
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> ( r1_tarski(B,D)
=> r1_tarski(C,D) ) ) ) ) ) ).
fof(d11_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_prob_1(B,u1_struct_0(A))
& v1_topgen_4(B,A)
& v3_topgen_4(B,u1_struct_0(A))
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> ( B = k5_topgen_4(A)
<=> ! [C] :
( ( v1_prob_1(C,u1_struct_0(A))
& v1_topgen_4(C,A)
& v3_topgen_4(C,u1_struct_0(A))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> r1_tarski(B,C) ) ) ) ) ).
fof(t68_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_tops_2(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> r1_tarski(B,k5_topgen_4(A)) ) ) ).
fof(t69_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_tops_2(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> r1_tarski(B,k5_topgen_4(A)) ) ) ).
fof(t70_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> k5_topgen_4(A) = k11_prob_1(u1_struct_0(A),k1_openlatt(A)) ) ).
fof(d12_topgen_4,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)))
=> ( v8_topgen_4(B,A)
<=> r2_hidden(B,k5_topgen_4(A)) ) ) ) ).
fof(dt_k1_topgen_4,axiom,
! [A] :
( l1_struct_0(A)
=> m1_subset_1(k1_topgen_4(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k2_topgen_4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> m1_subset_1(k2_topgen_4(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(commutativity_k2_topgen_4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k2_topgen_4(A,B,C) = k2_topgen_4(A,C,B) ) ).
fof(redefinition_k2_topgen_4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k2_topgen_4(A,B,C) = k3_setfam_1(B,C) ) ).
fof(dt_k3_topgen_4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> m1_subset_1(k3_topgen_4(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(commutativity_k3_topgen_4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k3_topgen_4(A,B,C) = k3_topgen_4(A,C,B) ) ).
fof(redefinition_k3_topgen_4,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k3_topgen_4(A,B,C) = k2_setfam_1(B,C) ) ).
fof(dt_k4_topgen_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k4_topgen_4(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k5_topgen_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_prob_1(k5_topgen_4(A),u1_struct_0(A))
& v1_topgen_4(k5_topgen_4(A),A)
& v3_topgen_4(k5_topgen_4(A),u1_struct_0(A))
& m1_subset_1(k5_topgen_4(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ).
fof(t2_topgen_4,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)))
=> k3_topgen_1(A,B) = a_2_0_topgen_4(A,B) ) ) ).
fof(s1_topgen_4,axiom,
v1_card_4(a_0_0_topgen_4) ).
fof(fraenkel_a_2_0_topgen_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_topgen_4(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r2_hidden(D,k6_pre_topc(B,k1_topgen_1(u1_struct_0(B),C,k1_struct_0(B,D)))) ) ) ) ).
fof(fraenkel_a_0_0_topgen_4,axiom,
! [A] :
( r2_hidden(A,a_0_0_topgen_4)
<=> ? [B] :
( m1_subset_1(B,k3_numbers)
& A = k1_tarski(B)
& p1_s1_topgen_4(B) ) ) ).
%------------------------------------------------------------------------------