SET007 Axioms: SET007+883.ax
%------------------------------------------------------------------------------
% File : SET007+883 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On constructing topological spaces and Sorgenfrey line
% 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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 78 ( 9 unt; 0 def)
% Number of atoms : 448 ( 79 equ)
% Maximal formula atoms : 28 ( 5 avg)
% Number of connectives : 448 ( 78 ~; 3 |; 195 &)
% ( 22 <=>; 150 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 43 ( 42 usr; 0 prp; 1-3 aty)
% Number of functors : 67 ( 67 usr; 14 con; 0-4 aty)
% Number of variables : 199 ( 188 !; 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_topgen_3,axiom,
( ~ v1_xboole_0(k4_numbers)
& ~ v1_finset_1(k4_numbers)
& v1_membered(k4_numbers)
& v2_membered(k4_numbers)
& v3_membered(k4_numbers)
& v4_membered(k4_numbers) ) ).
fof(fc2_topgen_3,axiom,
( ~ v1_xboole_0(k3_numbers)
& ~ v1_finset_1(k3_numbers)
& v1_membered(k3_numbers)
& v2_membered(k3_numbers)
& v3_membered(k3_numbers) ) ).
fof(fc3_topgen_3,axiom,
( ~ v1_xboole_0(k1_numbers)
& ~ v1_finset_1(k1_numbers)
& v1_membered(k1_numbers)
& v2_membered(k1_numbers) ) ).
fof(fc4_topgen_3,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ( ~ v1_xboole_0(k1_zfmisc_1(A))
& ~ v1_finset_1(k1_zfmisc_1(A)) ) ) ).
fof(fc5_topgen_3,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( ~ v1_xboole_0(k3_card_2(np__2,A))
& v1_ordinal1(k3_card_2(np__2,A))
& v2_ordinal1(k3_card_2(np__2,A))
& v3_ordinal1(k3_card_2(np__2,A))
& ~ v1_finset_1(k3_card_2(np__2,A))
& v1_card_1(k3_card_2(np__2,A)) ) ) ).
fof(rc1_topgen_3,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& ~ v1_finset_1(B) ) ) ).
fof(fc6_topgen_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k5_topgen_3(A))
& v1_pre_topc(k5_topgen_3(A))
& v2_pre_topc(k5_topgen_3(A)) ) ) ).
fof(fc7_topgen_3,axiom,
! [A,B] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k6_topgen_3(A,B))
& v1_pre_topc(k6_topgen_3(A,B))
& v2_pre_topc(k6_topgen_3(A,B)) ) ) ).
fof(fc8_topgen_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_tex_2(B,k1_zfmisc_1(A))
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ~ v1_xboole_0(k3_subset_1(A,B)) ) ).
fof(fc9_topgen_3,axiom,
! [A,B] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k7_topgen_3(A,B))
& v1_pre_topc(k7_topgen_3(A,B))
& v2_pre_topc(k7_topgen_3(A,B)) ) ) ).
fof(d1_topgen_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v1_topgen_3(B,A)
<=> ! [C,D,E] :
~ ( r2_hidden(D,B)
& r2_hidden(E,B)
& r2_hidden(C,k3_xboole_0(D,E))
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ~ ( r2_hidden(F,B)
& r2_hidden(C,F)
& r1_tarski(F,k3_xboole_0(D,E)) ) ) ) ) ) ).
fof(t1_topgen_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
<=> ! [C] :
~ ( r2_hidden(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( r2_hidden(D,B)
& r2_hidden(C,D) ) ) ) ) ) ).
fof(t2_topgen_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( v1_topgen_3(B,A)
& v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( l1_pre_topc(C)
=> ( ( u1_struct_0(C) = A
& u1_pre_topc(C) = k1_cantor_1(A,B) )
=> ( v2_pre_topc(C)
& l1_pre_topc(C)
& m1_cantor_1(B,C) ) ) ) ) ) ).
fof(t3_topgen_3,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ~ ( r1_tarski(k2_relat_1(B),k1_pcomps_1(k1_pcomps_1(A)))
& ! [C,D] :
( ( r2_hidden(C,A)
& r2_hidden(D,k1_funct_1(B,C)) )
=> r2_hidden(C,D) )
& ! [C,D,E] :
~ ( r2_hidden(C,E)
& r2_hidden(E,k1_funct_1(B,D))
& r2_hidden(D,A)
& ! [F] :
~ ( r2_hidden(F,k1_funct_1(B,C))
& r1_tarski(F,E) ) )
& ! [C,D,E] :
~ ( r2_hidden(C,A)
& r2_hidden(D,k1_funct_1(B,C))
& r2_hidden(E,k1_funct_1(B,C))
& ! [F] :
~ ( r2_hidden(F,k1_funct_1(B,C))
& r1_tarski(F,k3_xboole_0(D,E)) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ~ ( C = k3_card_3(B)
& ! [D] :
( l1_pre_topc(D)
=> ( ( u1_struct_0(D) = A
& u1_pre_topc(D) = k1_cantor_1(A,C) )
=> ( v2_pre_topc(D)
& l1_pre_topc(D)
& ! [E] :
( ( ~ v3_struct_0(E)
& v2_pre_topc(E)
& l1_pre_topc(E) )
=> ( E = D
=> m1_topgen_2(B,E) ) ) ) ) ) ) ) ) ) ).
fof(t4_topgen_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( r2_hidden(k12_arytm_3,B)
& r2_hidden(A,B)
& ! [C,D] :
( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k2_xboole_0(C,D),B) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r1_tarski(C,B)
=> r2_hidden(k8_setfam_1(A,C),B) ) ) )
=> ! [C] :
( l1_pre_topc(C)
=> ( ( u1_struct_0(C) = A
& u1_pre_topc(C) = k7_setfam_1(A,B) )
=> ( v2_pre_topc(C)
& l1_pre_topc(C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> ( v4_pre_topc(D,C)
<=> r2_hidden(D,B) ) ) ) ) ) ) ) ).
fof(t5_topgen_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ! [C] :
( ( v3_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
<=> ( v3_pre_topc(C,B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) ) )
=> g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) ) ) ) ).
fof(t6_topgen_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
<=> ( v4_pre_topc(C,B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) ) )
=> g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) ) ) ) ).
fof(t7_topgen_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_pcomps_1(A),k1_pcomps_1(A))
& m2_relset_1(B,k1_pcomps_1(A),k1_pcomps_1(A)) )
=> ( ( k1_funct_1(B,k12_arytm_3) = k12_arytm_3
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> r1_tarski(C,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C)) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,k4_subset_1(A,C,D)) = k4_subset_1(A,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C),k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,D)) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k8_funct_2(k1_pcomps_1(A),k1_pcomps_1(A),B,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C)) = k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C) ) )
=> ! [C] :
( l1_pre_topc(C)
=> ( ( u1_struct_0(C) = A
& u1_pre_topc(C) = k7_setfam_1(A,k1_topgen_3(k1_pcomps_1(A),A,B)) )
=> ( v2_pre_topc(C)
& l1_pre_topc(C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> k6_pre_topc(C,D) = k1_funct_1(B,D) ) ) ) ) ) ) ).
fof(t8_topgen_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ( u1_struct_0(A) = u1_struct_0(B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = D
=> k6_pre_topc(A,C) = k6_pre_topc(B,D) ) ) ) )
=> u1_pre_topc(A) = u1_pre_topc(B) ) ) ) ).
fof(t9_topgen_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_pcomps_1(A),k1_pcomps_1(A))
& m2_relset_1(B,k1_pcomps_1(A),k1_pcomps_1(A)) )
=> ( ( k1_funct_1(B,A) = A
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> r1_tarski(k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C),C) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,k5_subset_1(A,C,D)) = k5_subset_1(A,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C),k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,D)) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k8_funct_2(k1_pcomps_1(A),k1_pcomps_1(A),B,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C)) = k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C) ) )
=> ! [C] :
( l1_pre_topc(C)
=> ( ( u1_struct_0(C) = A
& u1_pre_topc(C) = k1_topgen_3(k1_pcomps_1(A),A,B) )
=> ( v2_pre_topc(C)
& l1_pre_topc(C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> k1_tops_1(C,D) = k1_funct_1(B,D) ) ) ) ) ) ) ).
fof(t10_topgen_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ( u1_struct_0(A) = u1_struct_0(B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( C = D
=> k1_tops_1(A,C) = k1_tops_1(B,D) ) ) ) )
=> u1_pre_topc(A) = u1_pre_topc(B) ) ) ) ).
fof(t11_topgen_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( v3_pre_topc(k1_rcomp_2(A,B),k2_topgen_3)
& m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ) ).
fof(t12_topgen_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( v3_pre_topc(k2_rcomp_1(A,B),k2_topgen_3)
& m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ) ).
fof(t13_topgen_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v3_pre_topc(k12_prob_1(A),k2_topgen_3)
& m1_subset_1(k12_prob_1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ).
fof(t14_topgen_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v3_pre_topc(k4_limfunc1(A),k2_topgen_3)
& m1_subset_1(k4_limfunc1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ).
fof(t15_topgen_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v3_pre_topc(k3_limfunc1(A),k2_topgen_3)
& m1_subset_1(k3_limfunc1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ).
fof(t16_topgen_3,axiom,
k1_card_1(k4_numbers) = k3_card_1(np__0) ).
fof(t17_topgen_3,axiom,
k1_card_1(k3_numbers) = k3_card_1(np__0) ).
fof(t18_topgen_3,axiom,
! [A] :
( ( v4_taxonom2(A)
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,C)
& r1_tarski(k2_rcomp_1(C,D),B) ) ) ) ) )
=> v1_card_4(A) ) ).
fof(d3_topgen_3,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ( r1_topgen_3(A,B)
<=> ( r2_hidden(B,A)
& ? [C] :
( v1_xreal_0(C)
& ~ r1_xreal_0(B,C)
& r1_xboole_0(k2_rcomp_1(C,B),A) ) ) ) ) ).
fof(d4_topgen_3,axiom,
k3_topgen_3 = k1_card_1(k1_numbers) ).
fof(d5_topgen_3,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( C = k4_topgen_3(A,B)
<=> ! [D] :
( v4_ordinal2(D)
=> ( ( r2_hidden(D,A)
& k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_newton(B,D) )
| ( ~ r2_hidden(D,A)
& k2_seq_1(k5_numbers,k1_numbers,C,D) = np__0 ) ) ) ) ) ) ).
fof(t21_topgen_3,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(np__1,B)
& ~ v1_series_1(k4_topgen_3(A,B)) ) ) ).
fof(t22_topgen_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& k2_series_1(k1_seqm_3(k2_prepower(A),B)) != k7_xcmplx_0(k2_newton(A,B),k6_xcmplx_0(np__1,A)) ) ) ) ).
fof(t23_topgen_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_series_1(k1_seqm_3(k2_prepower(k7_xcmplx_0(np__1,np__2)),k1_nat_1(A,np__1))) = k2_newton(k7_xcmplx_0(np__1,np__2),A) ) ).
fof(t24_topgen_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& ~ r1_xreal_0(k2_series_1(k4_topgen_3(B,A)),k2_series_1(k2_prepower(A))) ) ) ).
fof(t25_topgen_3,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_series_1(k1_seqm_3(k4_topgen_3(A,k7_xcmplx_0(np__1,np__2)),k1_nat_1(B,np__1))),k2_newton(k7_xcmplx_0(np__1,np__2),B)) ) ).
fof(t26_topgen_3,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
=> ! [B] :
( v4_ordinal2(B)
=> ~ r1_xreal_0(k2_series_1(k4_topgen_3(A,k7_xcmplx_0(np__1,np__2))),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k4_topgen_3(A,k7_xcmplx_0(np__1,np__2))),B)) ) ) ).
fof(t27_topgen_3,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
=> ! [B] :
( ( ~ v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k5_numbers)) )
=> ( k2_series_1(k4_topgen_3(A,k7_xcmplx_0(np__1,np__2))) = k2_series_1(k4_topgen_3(B,k7_xcmplx_0(np__1,np__2)))
=> A = B ) ) ) ).
fof(t28_topgen_3,axiom,
! [A] :
( v1_card_4(A)
=> v1_card_4(k5_finsub_1(A)) ) ).
fof(t29_topgen_3,axiom,
k3_topgen_3 = k3_card_2(np__2,k3_card_1(np__0)) ).
fof(t30_topgen_3,axiom,
r2_hidden(k3_card_1(np__0),k3_topgen_3) ).
fof(t32_topgen_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(k1_numbers)))
=> ~ ( r2_hidden(k1_card_1(A),k3_topgen_3)
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_rat_1(C)
=> ~ ( ~ r1_xreal_0(C,B)
& ~ r2_hidden(k1_rcomp_2(B,C),k1_cantor_1(k1_numbers,A)) ) ) ) ) ) ).
fof(t33_topgen_3,axiom,
k2_waybel23(k2_topgen_3) = k3_topgen_3 ).
fof(d6_topgen_3,axiom,
! [A,B] :
( ( v1_pre_topc(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( B = k5_topgen_3(A)
<=> ( u1_struct_0(B) = A
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v4_pre_topc(C,B)
<=> ( v1_finset_1(C)
| C = A ) ) ) ) ) ) ).
fof(t34_topgen_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_topgen_3(A))))
=> ( v3_pre_topc(B,k5_topgen_3(A))
<=> ( B = k12_arytm_3
| v1_finset_1(k3_subset_1(u1_struct_0(k5_topgen_3(A)),B)) ) ) ) ).
fof(t35_topgen_3,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ~ ( v1_finset_1(B)
& v1_finset_1(k3_subset_1(A,B)) ) ) ) ).
fof(t36_topgen_3,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_pre_topc(B,k5_topgen_3(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_topgen_3(A)))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v3_pre_topc(C,k5_topgen_3(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_3(A)))) )
=> ~ r2_subset_1(B,C) ) ) ) ).
fof(d7_topgen_3,axiom,
! [A,B,C] :
( ( v1_pre_topc(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( C = k6_topgen_3(A,B)
<=> ( u1_struct_0(C) = A
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> k6_pre_topc(C,D) = k1_cqc_lang(D,k12_arytm_3,D,k2_xboole_0(D,k3_xboole_0(k1_tarski(B),A))) ) ) ) ) ).
fof(t37_topgen_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
=> k6_pre_topc(k6_topgen_3(A,B),C) = k2_xboole_0(C,k6_domain_1(A,B)) ) ) ) ).
fof(t38_topgen_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
=> ( v4_pre_topc(C,k6_topgen_3(A,B))
<=> r2_hidden(B,C) ) ) ) ) ).
fof(t39_topgen_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_tex_2(C,k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
=> ( v3_pre_topc(C,k6_topgen_3(A,B))
<=> ~ r2_hidden(B,C) ) ) ) ) ).
fof(t40_topgen_3,axiom,
! [A,B,C] :
( r2_hidden(B,A)
=> ( ( v4_pre_topc(k1_tarski(C),k6_topgen_3(A,B))
& m1_subset_1(k1_tarski(C),k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
<=> C = B ) ) ).
fof(t41_topgen_3,axiom,
! [A,B,C] :
( r2_xboole_0(k1_tarski(B),A)
=> ( ( v3_pre_topc(k1_tarski(C),k6_topgen_3(A,B))
& m1_subset_1(k1_tarski(C),k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
<=> ( r2_hidden(C,A)
& C != B ) ) ) ).
fof(d8_topgen_3,axiom,
! [A,B,C] :
( ( v1_pre_topc(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( C = k7_topgen_3(A,B)
<=> ( u1_struct_0(C) = A
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> k1_tops_1(C,D) = k1_cqc_lang(D,A,D,k3_xboole_0(D,B)) ) ) ) ) ).
fof(t42_topgen_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_tex_2(C,k1_zfmisc_1(u1_struct_0(k7_topgen_3(A,B))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k7_topgen_3(A,B)))) )
=> k1_tops_1(k7_topgen_3(A,B),C) = k3_xboole_0(C,B) ) ) ).
fof(t43_topgen_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_tex_2(C,k1_zfmisc_1(u1_struct_0(k7_topgen_3(A,B))))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k7_topgen_3(A,B)))) )
=> ( v3_pre_topc(C,k7_topgen_3(A,B))
<=> r1_tarski(C,B) ) ) ) ).
fof(t44_topgen_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> u1_pre_topc(k7_topgen_3(A,B)) = k2_xboole_0(k1_tarski(A),k1_pcomps_1(B)) ) ).
fof(t45_topgen_3,axiom,
! [A] : k2_tex_1(A) = k7_topgen_3(A,k12_arytm_3) ).
fof(t46_topgen_3,axiom,
! [A] : k2_pcomps_1(A) = k7_topgen_3(A,A) ).
fof(s1_topgen_3,axiom,
( ( p1_s1_topgen_3(k12_arytm_3)
& p1_s1_topgen_3(f1_s1_topgen_3)
& ! [A,B] :
( ( p1_s1_topgen_3(A)
& p1_s1_topgen_3(B) )
=> p1_s1_topgen_3(k2_xboole_0(A,B)) )
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(f1_s1_topgen_3)))
=> ( ! [B] :
( r2_hidden(B,A)
=> p1_s1_topgen_3(B) )
=> p1_s1_topgen_3(k8_setfam_1(f1_s1_topgen_3,A)) ) ) )
=> ? [A] :
( v1_pre_topc(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& u1_struct_0(A) = f1_s1_topgen_3
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(B,A)
<=> p1_s1_topgen_3(B) ) ) ) ) ).
fof(s2_topgen_3,axiom,
( ( f2_s2_topgen_3(k12_arytm_3) = k12_arytm_3
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s2_topgen_3))
=> ( r1_tarski(A,f2_s2_topgen_3(A))
& r1_tarski(f2_s2_topgen_3(A),f1_s2_topgen_3) ) )
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s2_topgen_3))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(f1_s2_topgen_3))
=> f2_s2_topgen_3(k4_subset_1(f1_s2_topgen_3,A,B)) = k2_xboole_0(f2_s2_topgen_3(A),f2_s2_topgen_3(B)) ) )
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s2_topgen_3))
=> f2_s2_topgen_3(f2_s2_topgen_3(A)) = f2_s2_topgen_3(A) ) )
=> ? [A] :
( v1_pre_topc(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& u1_struct_0(A) = f1_s2_topgen_3
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k6_pre_topc(A,B) = f2_s2_topgen_3(B) ) ) ) ).
fof(s3_topgen_3,axiom,
( ( f2_s3_topgen_3(f1_s3_topgen_3) = f1_s3_topgen_3
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s3_topgen_3))
=> r1_tarski(f2_s3_topgen_3(A),A) )
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s3_topgen_3))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(f1_s3_topgen_3))
=> f2_s3_topgen_3(k5_subset_1(f1_s3_topgen_3,A,B)) = k3_xboole_0(f2_s3_topgen_3(A),f2_s3_topgen_3(B)) ) )
& ! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s3_topgen_3))
=> f2_s3_topgen_3(f2_s3_topgen_3(A)) = f2_s3_topgen_3(A) ) )
=> ? [A] :
( v1_pre_topc(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& u1_struct_0(A) = f1_s3_topgen_3
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_tops_1(A,B) = f2_s3_topgen_3(B) ) ) ) ).
fof(dt_k1_topgen_3,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_pcomps_1(B))))
=> m1_subset_1(k1_topgen_3(A,B,C),k1_zfmisc_1(k1_zfmisc_1(B))) ) ).
fof(redefinition_k1_topgen_3,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_pcomps_1(B))))
=> k1_topgen_3(A,B,C) = k2_relat_1(C) ) ).
fof(dt_k2_topgen_3,axiom,
( ~ v3_struct_0(k2_topgen_3)
& v1_pre_topc(k2_topgen_3)
& v2_pre_topc(k2_topgen_3)
& l1_pre_topc(k2_topgen_3) ) ).
fof(dt_k3_topgen_3,axiom,
( ~ v1_finset_1(k3_topgen_3)
& v1_card_1(k3_topgen_3) ) ).
fof(dt_k4_topgen_3,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ( v1_funct_1(k4_topgen_3(A,B))
& v1_funct_2(k4_topgen_3(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k4_topgen_3(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k5_topgen_3,axiom,
! [A] :
( v1_pre_topc(k5_topgen_3(A))
& v2_pre_topc(k5_topgen_3(A))
& l1_pre_topc(k5_topgen_3(A)) ) ).
fof(dt_k6_topgen_3,axiom,
! [A,B] :
( v1_pre_topc(k6_topgen_3(A,B))
& v2_pre_topc(k6_topgen_3(A,B))
& l1_pre_topc(k6_topgen_3(A,B)) ) ).
fof(dt_k7_topgen_3,axiom,
! [A,B] :
( v1_pre_topc(k7_topgen_3(A,B))
& v2_pre_topc(k7_topgen_3(A,B))
& l1_pre_topc(k7_topgen_3(A,B)) ) ).
fof(d2_topgen_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( A = k2_topgen_3
<=> ( u1_struct_0(A) = k1_numbers
& ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_numbers)))
& u1_pre_topc(A) = k1_cantor_1(k1_numbers,B)
& B = a_0_0_topgen_3 ) ) ) ) ).
fof(t19_topgen_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> v1_card_4(a_1_0_topgen_3(A)) ) ).
fof(t20_topgen_3,axiom,
k1_card_1(a_0_0_topgen_3) = k3_topgen_3 ).
fof(t31_topgen_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(k1_numbers)))
=> ( r2_hidden(k1_card_1(A),k3_topgen_3)
=> r2_hidden(k1_card_1(a_1_1_topgen_3(A)),k3_topgen_3) ) ) ).
fof(fraenkel_a_0_0_topgen_3,axiom,
! [A] :
( r2_hidden(A,a_0_0_topgen_3)
<=> ? [B,C] :
( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& A = k1_rcomp_2(B,C)
& ~ r1_xreal_0(C,B)
& v1_rat_1(C) ) ) ).
fof(fraenkel_a_1_0_topgen_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( r2_hidden(A,a_1_0_topgen_3(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& r1_topgen_3(B,C) ) ) ) ).
fof(fraenkel_a_1_1_topgen_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_numbers)))
=> ( r2_hidden(A,a_1_1_topgen_3(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& ? [D] :
( r2_hidden(D,k1_cantor_1(k1_numbers,B))
& r1_topgen_3(D,C) ) ) ) ) ).
%------------------------------------------------------------------------------