SET007 Axioms: SET007+882.ax
%------------------------------------------------------------------------------
% File : SET007+882 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the characteristic and weight of a topological space
% 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_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 70 ( 3 unt; 0 def)
% Number of atoms : 403 ( 45 equ)
% Maximal formula atoms : 19 ( 5 avg)
% Number of connectives : 402 ( 69 ~; 1 |; 168 &)
% ( 18 <=>; 146 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-3 aty)
% Number of functors : 46 ( 46 usr; 1 con; 0-4 aty)
% Number of variables : 202 ( 181 !; 21 ?)
% 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(rc1_topgen_2,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))))
& ~ v1_xboole_0(B)
& v1_tops_2(B,A) ) ) ).
fof(cc1_topgen_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v6_group_1(A)
=> v1_topgen_2(A) ) ) ).
fof(cc2_topgen_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ~ v1_topgen_2(A)
=> ( ~ v3_struct_0(A)
& ~ v6_group_1(A)
& ~ v3_realset2(A) ) ) ) ).
fof(rc2_topgen_2,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v6_group_1(A)
& v1_topgen_2(A) ) ).
fof(rc3_topgen_2,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v6_group_1(A)
& ~ v3_realset2(A)
& ~ v1_topgen_2(A) ) ).
fof(fc1_topgen_2,axiom,
! [A,B] :
( v1_pre_topc(k5_topgen_2(A,B))
& v2_pre_topc(k5_topgen_2(A,B)) ) ).
fof(fc2_topgen_2,axiom,
! [A,B] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k5_topgen_2(A,B))
& v1_pre_topc(k5_topgen_2(A,B))
& v2_pre_topc(k5_topgen_2(A,B)) ) ) ).
fof(t2_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> m1_yellow_8(k1_funct_1(B,C),A,C) )
=> m1_cantor_1(k3_card_3(B),A) ) ) ) ).
fof(d1_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_card_1(C)
=> ( C = k1_topgen_2(A,B)
<=> ( ? [D] :
( m1_yellow_8(D,A,B)
& C = k1_card_1(D) )
& ! [D] :
( m1_yellow_8(D,A,B)
=> r1_tarski(C,k1_card_1(D)) ) ) ) ) ) ) ).
fof(t3_topgen_2,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> v1_card_1(B) )
=> v1_card_1(k3_tarski(A)) ) ).
fof(d2_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( v1_card_1(B)
=> ( B = k2_topgen_2(A)
<=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_tarski(k1_topgen_2(A,C),B) )
& ! [C] :
( v1_card_1(C)
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_tarski(k1_topgen_2(A,D),C) )
=> r1_tarski(B,C) ) ) ) ) ) ) ).
fof(t5_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_tarski(k1_topgen_2(A,B),k2_topgen_2(A)) ) ) ).
fof(t6_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_frechet(A)
<=> r1_tarski(k2_topgen_2(A),k3_card_1(np__0)) ) ) ).
fof(d3_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ( m1_topgen_2(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> m1_yellow_8(k1_funct_1(B,C),A,C) ) ) ) ) ).
fof(t7_topgen_2,axiom,
$true ).
fof(t8_topgen_2,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))
=> ! [D] :
( m1_yellow_8(D,A,B)
=> ! [E] :
( m1_yellow_8(E,A,C)
=> ! [F] :
~ ( r2_hidden(B,F)
& r2_hidden(F,E)
& ! [G] :
( ( v3_pre_topc(G,A)
& m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r2_hidden(G,D)
& r1_tarski(G,F) ) ) ) ) ) ) ) ) ).
fof(t9_topgen_2,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_yellow_8(C,A,B)
=> ! [D,E] :
~ ( r2_hidden(D,C)
& r2_hidden(E,C)
& ! [F] :
( ( v3_pre_topc(F,A)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r2_hidden(F,C)
& r1_tarski(F,k3_xboole_0(D,E)) ) ) ) ) ) ) ).
fof(t10_topgen_2,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,k6_pre_topc(A,B))
<=> ! [D] :
( m1_yellow_8(D,A,C)
=> ! [E] :
~ ( r2_hidden(E,D)
& r1_xboole_0(E,B) ) ) ) ) ) ) ).
fof(t11_topgen_2,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,k6_pre_topc(A,B))
<=> ? [D] :
( m1_yellow_8(D,A,C)
& ! [E] :
~ ( r2_hidden(E,D)
& r1_xboole_0(E,B) ) ) ) ) ) ) ).
fof(t12_topgen_2,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)))) )
=> ? [C] :
( v1_tops_2(C,A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& r1_tarski(C,B)
& k5_setfam_1(u1_struct_0(A),C) = k5_setfam_1(u1_struct_0(A),B)
& r1_tarski(k1_card_1(C),k2_waybel23(A)) ) ) ) ).
fof(d4_topgen_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v1_topgen_2(A)
<=> v1_finset_1(k2_waybel23(A)) ) ) ).
fof(t13_topgen_2,axiom,
! [A] : k1_card_1(k3_pua2mss1(A)) = k1_card_1(A) ).
fof(t14_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> ( m1_cantor_1(k3_pua2mss1(u1_struct_0(A)),A)
& ! [B] :
( m1_cantor_1(B,A)
=> r1_tarski(k3_pua2mss1(u1_struct_0(A)),B) ) ) ) ).
fof(t15_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_tdlat_3(A)
& l1_pre_topc(A) )
=> k2_waybel23(A) = k1_card_1(u1_struct_0(A)) ) ).
fof(t16_topgen_2,axiom,
$true ).
fof(t17_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_topgen_2(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_cantor_1(B,A)
& ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_pre_topc(A))
& m2_relset_1(C,u1_struct_0(A),u1_pre_topc(A))
& B = k5_relset_1(u1_struct_0(A),u1_pre_topc(A),C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,k8_funct_2(u1_struct_0(A),u1_pre_topc(A),C,D))
& ! [E] :
( ( v3_pre_topc(E,A)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_hidden(D,E)
=> r1_tarski(k8_funct_2(u1_struct_0(A),u1_pre_topc(A),C,D),E) ) ) ) ) ) ) ) ).
fof(t18_topgen_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_cantor_1(B,A)
=> ! [C] :
( m1_cantor_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_pre_topc(A))
& m2_relset_1(D,u1_struct_0(A),u1_pre_topc(A)) )
=> ( ( B = k5_relset_1(u1_struct_0(A),u1_pre_topc(A),D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,k1_funct_1(D,E))
& ! [F] :
( ( v3_pre_topc(F,A)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_hidden(E,F)
=> r1_tarski(k1_funct_1(D,E),F) ) ) ) ) )
=> r1_tarski(B,C) ) ) ) ) ) ).
fof(t19_topgen_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_cantor_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_pre_topc(A))
& m2_relset_1(C,u1_struct_0(A),u1_pre_topc(A)) )
=> ( ( B = k5_relset_1(u1_struct_0(A),u1_pre_topc(A),C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,k1_funct_1(C,D))
& ! [E] :
( ( v3_pre_topc(E,A)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_hidden(D,E)
=> r1_tarski(k1_funct_1(C,D),E) ) ) ) ) )
=> k2_waybel23(A) = k1_card_1(B) ) ) ) ) ).
fof(t20_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_cantor_1(B,A)
=> ? [C] :
( m1_cantor_1(C,A)
& r1_tarski(C,B)
& k1_card_1(C) = k2_waybel23(A) ) ) ) ).
fof(t21_topgen_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
=> ( v3_pre_topc(C,k5_topgen_2(A,B))
<=> ~ ( r2_hidden(B,C)
& ~ v1_finset_1(k3_subset_1(u1_struct_0(k5_topgen_2(A,B)),C)) ) ) ) ).
fof(t22_topgen_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
=> ( v4_pre_topc(C,k5_topgen_2(A,B))
<=> ~ ( r2_hidden(B,A)
& ~ r2_hidden(B,C)
& ~ v1_finset_1(C) ) ) ) ).
fof(t23_topgen_2,axiom,
! [A,B,C] :
( r2_hidden(C,A)
=> ( v4_pre_topc(k1_tarski(C),k5_topgen_2(A,B))
& m1_subset_1(k1_tarski(C),k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B)))) ) ) ).
fof(t24_topgen_2,axiom,
! [A,B,C] :
( r2_hidden(C,A)
=> ( C = B
| ( v3_pre_topc(k1_tarski(C),k5_topgen_2(A,B))
& m1_subset_1(k1_tarski(C),k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B)))) ) ) ) ).
fof(t25_topgen_2,axiom,
! [A,B] :
( ~ v1_finset_1(A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
=> ~ ( C = k1_tarski(B)
& v3_pre_topc(C,k5_topgen_2(A,B)) ) ) ) ).
fof(t26_topgen_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
=> ( v1_finset_1(C)
=> k6_pre_topc(k5_topgen_2(A,B),C) = C ) ) ).
fof(t27_topgen_2,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)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( v4_pre_topc(k4_subset_1(u1_struct_0(A),B,k1_struct_0(A,C)),A)
=> k6_pre_topc(A,B) = k4_subset_1(u1_struct_0(A),B,k1_struct_0(A,C)) ) ) ) ) ) ).
fof(t28_topgen_2,axiom,
! [A,B] :
( r2_hidden(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
=> ( ~ v1_finset_1(C)
=> k6_pre_topc(k5_topgen_2(A,B),C) = k2_xboole_0(C,k1_tarski(B)) ) ) ) ).
fof(t29_topgen_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
=> ( v1_finset_1(k3_subset_1(u1_struct_0(k5_topgen_2(A,B)),C))
=> k1_tops_1(k5_topgen_2(A,B),C) = C ) ) ).
fof(t30_topgen_2,axiom,
! [A,B] :
( r2_hidden(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_2(A,B))))
=> ( ~ v1_finset_1(k3_subset_1(u1_struct_0(k5_topgen_2(A,B)),C))
=> k1_tops_1(k5_topgen_2(A,B),C) = k4_xboole_0(C,k1_tarski(B)) ) ) ) ).
fof(t32_topgen_2,axiom,
! [A] :
( ~ v1_finset_1(A)
=> k1_card_1(k5_finsub_1(A)) = k1_card_1(A) ) ).
fof(t35_topgen_2,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B,C] :
( m1_cantor_1(C,k5_topgen_2(A,B))
=> r1_tarski(k1_card_1(A),k1_card_1(C)) ) ) ).
fof(t36_topgen_2,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] : k2_waybel23(k5_topgen_2(A,B)) = k1_card_1(A) ) ).
fof(t40_topgen_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_pcomps_1(A))))
=> ~ ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r2_hidden(C,B)
=> ( v2_pre_topc(g1_pre_topc(A,C))
& l1_pre_topc(g1_pre_topc(A,C)) ) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ~ ( C = k8_setfam_1(k1_pcomps_1(A),B)
& v2_pre_topc(g1_pre_topc(A,C))
& l1_pre_topc(g1_pre_topc(A,C))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r2_hidden(D,B)
=> m2_yellow_9(g1_pre_topc(A,D),g1_pre_topc(A,C)) ) )
& ! [D] :
( ( v2_pre_topc(D)
& l1_pre_topc(D) )
=> ( ( u1_struct_0(D) = A
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r2_hidden(E,B)
=> m2_yellow_9(g1_pre_topc(A,E),D) ) ) )
=> m2_yellow_9(g1_pre_topc(A,C),D) ) ) ) ) ) ) ).
fof(t41_topgen_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_pcomps_1(A))))
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
& C = k1_cantor_1(A,k2_cantor_1(A,k5_setfam_1(k1_pcomps_1(A),B)))
& v2_pre_topc(g1_pre_topc(A,C))
& l1_pre_topc(g1_pre_topc(A,C))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r2_hidden(D,B)
=> m2_yellow_9(g1_pre_topc(A,C),g1_pre_topc(A,D)) ) )
& ! [D] :
( ( v2_pre_topc(D)
& l1_pre_topc(D) )
=> ( ( u1_struct_0(D) = A
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r2_hidden(E,B)
=> m2_yellow_9(D,g1_pre_topc(A,E)) ) ) )
=> m2_yellow_9(D,g1_pre_topc(A,C)) ) ) ) ) ).
fof(dt_m1_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_topgen_2(B,A)
=> m1_pboole(B,u1_struct_0(A)) ) ) ).
fof(existence_m1_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] : m1_topgen_2(B,A) ) ).
fof(dt_k1_topgen_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> v1_card_1(k1_topgen_2(A,B)) ) ).
fof(dt_k2_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> v1_card_1(k2_topgen_2(A)) ) ).
fof(dt_k3_topgen_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_topgen_2(B,A) )
=> m1_cantor_1(k3_topgen_2(A,B),A) ) ).
fof(redefinition_k3_topgen_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_topgen_2(B,A) )
=> k3_topgen_2(A,B) = k3_card_3(B) ) ).
fof(dt_k4_topgen_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_topgen_2(B,A)
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_yellow_8(k4_topgen_2(A,B,C),A,C) ) ).
fof(redefinition_k4_topgen_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_topgen_2(B,A)
& m1_subset_1(C,u1_struct_0(A)) )
=> k4_topgen_2(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k5_topgen_2,axiom,
! [A,B] :
( v1_pre_topc(k5_topgen_2(A,B))
& l1_pre_topc(k5_topgen_2(A,B)) ) ).
fof(t1_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_cantor_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> m1_yellow_8(a_3_0_topgen_2(A,B,C),A,C) ) ) ) ).
fof(t4_topgen_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> k2_topgen_2(A) = k3_tarski(a_1_0_topgen_2(A)) ) ).
fof(d5_topgen_2,axiom,
! [A,B,C] :
( ( v1_pre_topc(C)
& l1_pre_topc(C) )
=> ( C = k5_topgen_2(A,B)
<=> ( u1_struct_0(C) = A
& u1_pre_topc(C) = k2_xboole_0(a_2_0_topgen_2(A,B),a_1_1_topgen_2(A)) ) ) ) ).
fof(t31_topgen_2,axiom,
! [A,B] :
? [C] :
( m1_cantor_1(C,k5_topgen_2(A,B))
& C = k2_xboole_0(k4_xboole_0(k3_pua2mss1(A),k1_tarski(k1_tarski(B))),a_1_1_topgen_2(A)) ) ).
fof(t33_topgen_2,axiom,
! [A] :
( ~ v1_finset_1(A)
=> k1_card_1(a_1_2_topgen_2(A)) = k1_card_1(A) ) ).
fof(t34_topgen_2,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B,C] :
( m1_cantor_1(C,k5_topgen_2(A,B))
=> ( C = k2_xboole_0(k4_xboole_0(k3_pua2mss1(A),k1_tarski(k1_tarski(B))),a_1_2_topgen_2(A))
=> k1_card_1(C) = k1_card_1(A) ) ) ) ).
fof(t37_topgen_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
? [C] :
( m2_cantor_1(C,k5_topgen_2(A,B))
& C = k2_xboole_0(k4_xboole_0(k3_pua2mss1(A),k1_tarski(k1_tarski(B))),a_1_3_topgen_2(A)) ) ) ).
fof(t38_topgen_2,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] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
=> ( ! [D] :
( r2_hidden(D,C)
=> v1_finset_1(k4_xboole_0(B,D)) )
=> k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k2_xboole_0(k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)),k1_setfam_1(a_3_1_topgen_2(A,B,C))) ) ) ) ) ).
fof(t39_topgen_2,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))))
=> k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k2_xboole_0(k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)),k1_setfam_1(a_2_1_topgen_2(A,B))) ) ) ).
fof(fraenkel_a_3_0_topgen_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_cantor_1(C,B)
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_topgen_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& A = E
& r2_hidden(D,E)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_1_0_topgen_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ( r2_hidden(A,a_1_0_topgen_2(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = k1_topgen_2(B,C) ) ) ) ).
fof(fraenkel_a_2_0_topgen_2,axiom,
! [A,B,C] :
( r2_hidden(A,a_2_0_topgen_2(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& ~ r2_hidden(C,D) ) ) ).
fof(fraenkel_a_1_1_topgen_2,axiom,
! [A,B] :
( r2_hidden(A,a_1_1_topgen_2(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
& A = k3_subset_1(B,C)
& v1_finset_1(C) ) ) ).
fof(fraenkel_a_1_2_topgen_2,axiom,
! [A,B] :
( ~ v1_finset_1(B)
=> ( r2_hidden(A,a_1_2_topgen_2(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
& A = k3_subset_1(B,C)
& v1_finset_1(C) ) ) ) ).
fof(fraenkel_a_1_3_topgen_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_3_topgen_2(B))
<=> ? [C] :
( m1_subset_1(C,B)
& A = k3_subset_1(B,k6_domain_1(B,C)) ) ) ) ).
fof(fraenkel_a_3_1_topgen_2,axiom,
! [A,B,C,D] :
( ( v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(C))) )
=> ( r2_hidden(A,a_3_1_topgen_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
& A = k6_pre_topc(B,k5_setfam_1(u1_struct_0(B),E))
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_2_1_topgen_2,axiom,
! [A,B,C] :
( ( v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) )
=> ( r2_hidden(A,a_2_1_topgen_2(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
& A = k6_pre_topc(B,k5_setfam_1(u1_struct_0(B),D))
& r1_tarski(D,C)
& v1_finset_1(k6_subset_1(k1_zfmisc_1(u1_struct_0(B)),C,D)) ) ) ) ).
%------------------------------------------------------------------------------