SET007 Axioms: SET007+438.ax
%------------------------------------------------------------------------------
% File : SET007+438 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Dyadic Numbers and T_4 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 : urysohn1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 58 ( 14 unt; 0 def)
% Number of atoms : 336 ( 32 equ)
% Maximal formula atoms : 36 ( 5 avg)
% Number of connectives : 335 ( 57 ~; 4 |; 138 &)
% ( 16 <=>; 120 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 27 ( 25 usr; 1 prp; 0-4 aty)
% Number of functors : 35 ( 35 usr; 11 con; 0-4 aty)
% Number of variables : 122 ( 117 !; 5 ?)
% 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_urysohn1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k3_urysohn1(A))
& v1_membered(k3_urysohn1(A))
& v2_membered(k3_urysohn1(A)) ) ) ).
fof(fc2_urysohn1,axiom,
( ~ v1_xboole_0(k4_urysohn1)
& v1_membered(k4_urysohn1)
& v2_membered(k4_urysohn1) ) ).
fof(fc3_urysohn1,axiom,
( ~ v1_xboole_0(k5_urysohn1)
& v1_membered(k5_urysohn1)
& v2_membered(k5_urysohn1) ) ).
fof(d1_urysohn1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( A = k1_urysohn1
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(B,A)
<=> ~ r1_xreal_0(np__0,B) ) ) ) ) ).
fof(d2_urysohn1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( A = k2_urysohn1
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(B,A)
<=> ~ r1_xreal_0(B,np__1) ) ) ) ) ).
fof(d3_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( B = k3_urysohn1(A)
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,B)
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(np__0,D)
& r1_xreal_0(D,k1_card_4(np__2,A))
& C = k6_real_1(D,k1_card_4(np__2,A)) ) ) ) ) ) ) ).
fof(d4_urysohn1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( A = k4_urysohn1
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(B,A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r2_hidden(B,k3_urysohn1(C)) ) ) ) ) ) ).
fof(d5_urysohn1,axiom,
k5_urysohn1 = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k1_urysohn1,k4_urysohn1),k2_urysohn1) ).
fof(t1_urysohn1,axiom,
$true ).
fof(t2_urysohn1,axiom,
$true ).
fof(t3_urysohn1,axiom,
$true ).
fof(t4_urysohn1,axiom,
$true ).
fof(t5_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(B,k3_urysohn1(A))
=> ( r1_xreal_0(np__0,B)
& r1_xreal_0(B,np__1) ) ) ) ) ).
fof(t6_urysohn1,axiom,
k3_urysohn1(np__0) = k7_domain_1(k1_numbers,np__0,np__1) ).
fof(t7_urysohn1,axiom,
k3_urysohn1(np__1) = k8_domain_1(k1_numbers,np__0,k6_real_1(np__1,np__2),np__1) ).
fof(t8_urysohn1,axiom,
$true ).
fof(t9_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& k4_finseq_1(B) = k2_finseq_1(k1_nat_1(k1_card_4(np__2,A),np__1))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> k1_funct_1(B,C) = k6_real_1(k5_real_1(C,np__1),k1_card_4(np__2,A)) ) ) ) ) ).
fof(d6_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( B = k7_urysohn1(A)
<=> ( k4_finseq_1(B) = k2_finseq_1(k1_nat_1(k1_card_4(np__2,A),np__1))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> k1_funct_1(B,C) = k6_real_1(k5_real_1(C,np__1),k1_card_4(np__2,A)) ) ) ) ) ) ) ).
fof(t10_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k4_finseq_1(k7_urysohn1(A)) = k2_finseq_1(k1_nat_1(k1_card_4(np__2,A),np__1))
& k2_relat_1(k7_urysohn1(A)) = k3_urysohn1(A) ) ) ).
fof(t11_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k3_urysohn1(A),k3_urysohn1(k1_nat_1(A,np__1))) ) ).
fof(t12_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r2_hidden(np__0,k3_urysohn1(A))
& r2_hidden(np__1,k3_urysohn1(A)) ) ) ).
fof(t13_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,k1_card_4(np__2,A))
=> ( r1_xreal_0(B,np__0)
| r2_hidden(k6_real_1(k5_real_1(k2_nat_1(B,np__2),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k6_subset_1(k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)),k3_urysohn1(A))) ) ) ) ) ).
fof(t14_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__0,B)
=> ( r1_xreal_0(k1_card_4(np__2,A),B)
| r2_hidden(k6_real_1(k1_nat_1(k2_nat_1(B,np__2),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k6_subset_1(k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)),k3_urysohn1(A))) ) ) ) ) ).
fof(t15_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r2_hidden(k6_real_1(np__1,k1_card_4(np__2,k1_nat_1(A,np__1))),k6_subset_1(k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)),k3_urysohn1(A))) ) ).
fof(d7_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k8_urysohn1(A,B)
<=> B = k6_real_1(C,k1_card_4(np__2,A)) ) ) ) ) ).
fof(t16_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
=> ( B = k6_real_1(k8_urysohn1(A,B),k1_card_4(np__2,A))
& r1_xreal_0(np__0,k8_urysohn1(A,B))
& r1_xreal_0(k8_urysohn1(A,B),k1_card_4(np__2,A)) ) ) ) ).
fof(t17_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
=> ( ~ r1_xreal_0(B,k6_real_1(k5_real_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A)))
& ~ r1_xreal_0(k6_real_1(k1_nat_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A)),B) ) ) ) ).
fof(t18_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
=> ~ r1_xreal_0(k6_real_1(k1_nat_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A)),k6_real_1(k5_real_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A))) ) ) ).
fof(t19_urysohn1,axiom,
$true ).
fof(t20_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)))
=> ( ~ r2_hidden(B,k3_urysohn1(A))
=> ( r2_hidden(k6_real_1(k5_real_1(k8_urysohn1(k1_nat_1(A,np__1),B),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k3_urysohn1(A))
& r2_hidden(k6_real_1(k1_nat_1(k8_urysohn1(k1_nat_1(A,np__1),B),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k3_urysohn1(A)) ) ) ) ) ).
fof(t21_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k3_urysohn1(A))
=> ~ ( ~ r1_xreal_0(C,B)
& r1_xreal_0(k8_urysohn1(A,C),k8_urysohn1(A,B)) ) ) ) ) ).
fof(t22_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k3_urysohn1(A))
=> ( ~ r1_xreal_0(C,B)
=> ( r1_xreal_0(B,k6_real_1(k5_real_1(k8_urysohn1(A,C),np__1),k1_card_4(np__2,A)))
& r1_xreal_0(k6_real_1(k1_nat_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A)),C) ) ) ) ) ) ).
fof(t23_urysohn1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)))
=> ~ ( ~ r1_xreal_0(C,B)
& ~ r2_hidden(B,k3_urysohn1(A))
& ~ r2_hidden(C,k3_urysohn1(A))
& ~ r1_xreal_0(k6_real_1(k1_nat_1(k8_urysohn1(k1_nat_1(A,np__1),B),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k6_real_1(k5_real_1(k8_urysohn1(k1_nat_1(A,np__1),C),np__1),k1_card_4(np__2,k1_nat_1(A,np__1)))) ) ) ) ) ).
fof(d8_urysohn1,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_connsp_2(C,A,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& v3_pre_topc(D,A)
& r2_hidden(B,D)
& r1_tarski(D,C) ) ) ) ) ) ).
fof(t24_urysohn1,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)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_connsp_2(D,A,C)
=> ~ r1_tarski(D,B) ) ) ) ) ) ) ).
fof(t25_urysohn1,axiom,
$true ).
fof(t26_urysohn1,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,B)
=> m1_connsp_2(B,A,C) ) )
=> v3_pre_topc(B,A) ) ) ) ).
fof(d9_urysohn1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v1_urysohn1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& v3_pre_topc(E,A)
& r2_hidden(B,D)
& ~ r2_hidden(C,D)
& r2_hidden(C,E)
& ~ r2_hidden(B,E) ) ) ) ) ) ) ) ) ).
fof(t27_urysohn1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_urysohn1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v4_pre_topc(k1_struct_0(A,B),A) ) ) ) ).
fof(t28_urysohn1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v5_compts_1(A)
=> ! [B] :
( ( v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v3_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( B != k1_xboole_0
& r1_tarski(k6_pre_topc(A,B),C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( D != k1_xboole_0
& v3_pre_topc(D,A)
& r1_tarski(k6_pre_topc(A,B),D)
& r1_tarski(k6_pre_topc(A,D),C) ) ) ) ) ) ) ) ).
fof(t29_urysohn1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v4_compts_1(A)
<=> ! [B] :
( ( v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( ( v3_pre_topc(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r2_hidden(C,D)
& r1_tarski(k6_pre_topc(A,D),B) ) ) ) ) ) ) ) ).
fof(t30_urysohn1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v5_compts_1(A)
& v1_urysohn1(A) )
=> ! [B] :
( ( v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( B != k1_xboole_0
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( C != k1_xboole_0
& r1_tarski(k6_pre_topc(A,C),B) ) ) ) ) ) ) ).
fof(t31_urysohn1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v5_compts_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(B,A)
& v4_pre_topc(C,A)
& C != k1_xboole_0
& r1_tarski(C,B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& r1_tarski(C,D)
& r1_tarski(k6_pre_topc(A,D),B) ) ) ) ) ) ) ) ).
fof(d10_urysohn1,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_compts_1(A)
& v3_pre_topc(B,A)
& v3_pre_topc(C,A)
& r1_tarski(k6_pre_topc(A,B),C) )
=> ( B = k1_xboole_0
| ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_urysohn1(D,A,B,C)
<=> ( D != k1_xboole_0
& v3_pre_topc(D,A)
& r1_tarski(k6_pre_topc(A,B),D)
& r1_tarski(k6_pre_topc(A,D),C) ) ) ) ) ) ) ) ) ).
fof(t32_urysohn1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v5_compts_1(A)
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k1_xboole_0
| ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_urysohn1(D),k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(E,k3_urysohn1(D),k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(B,k1_funct_1(E,np__0))
& C = k4_xboole_0(k2_pre_topc(A),k1_funct_1(E,np__1))
& ! [F] :
( m2_subset_1(F,k1_numbers,k3_urysohn1(D))
=> ! [G] :
( m2_subset_1(G,k1_numbers,k3_urysohn1(D))
=> ( ~ r1_xreal_0(G,F)
=> ( v3_pre_topc(k6_urysohn1(A,k3_urysohn1(D),E,F),A)
& v3_pre_topc(k6_urysohn1(A,k3_urysohn1(D),E,G),A)
& r1_tarski(k6_pre_topc(A,k6_urysohn1(A,k3_urysohn1(D),E,F)),k6_urysohn1(A,k3_urysohn1(D),E,G)) ) ) ) )
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k3_urysohn1(k1_nat_1(D,np__1)),k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(F,k3_urysohn1(k1_nat_1(D,np__1)),k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(B,k1_funct_1(F,np__0))
& C = k4_xboole_0(k2_pre_topc(A),k1_funct_1(F,np__1))
& ! [G] :
( m2_subset_1(G,k1_numbers,k3_urysohn1(k1_nat_1(D,np__1)))
=> ! [H] :
( m2_subset_1(H,k1_numbers,k3_urysohn1(k1_nat_1(D,np__1)))
=> ! [I] :
( m2_subset_1(I,k1_numbers,k3_urysohn1(k1_nat_1(D,np__1)))
=> ( ~ r1_xreal_0(H,G)
=> ( v3_pre_topc(k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,G),A)
& v3_pre_topc(k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,H),A)
& r1_tarski(k6_pre_topc(A,k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,G)),k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,H))
& ( r2_hidden(I,k3_urysohn1(D))
=> k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,I) = k1_funct_1(E,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_m1_urysohn1,axiom,
! [A,B,C] :
( ( ~ 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(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_urysohn1(D,A,B,C)
=> m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m1_urysohn1,axiom,
! [A,B,C] :
( ( ~ 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(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [D] : m1_urysohn1(D,A,B,C) ) ).
fof(dt_k1_urysohn1,axiom,
m1_subset_1(k1_urysohn1,k1_zfmisc_1(k1_numbers)) ).
fof(dt_k2_urysohn1,axiom,
m1_subset_1(k2_urysohn1,k1_zfmisc_1(k1_numbers)) ).
fof(dt_k3_urysohn1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k3_urysohn1(A),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k4_urysohn1,axiom,
m1_subset_1(k4_urysohn1,k1_zfmisc_1(k1_numbers)) ).
fof(dt_k5_urysohn1,axiom,
m1_subset_1(k5_urysohn1,k1_zfmisc_1(k1_numbers)) ).
fof(dt_k6_urysohn1,axiom,
! [A,B,C,D] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers))
& v1_funct_1(C)
& v1_funct_2(C,B,k1_zfmisc_1(u1_struct_0(A)))
& m1_relset_1(C,B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(D,B) )
=> m1_subset_1(k6_urysohn1(A,B,C,D),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(redefinition_k6_urysohn1,axiom,
! [A,B,C,D] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers))
& v1_funct_1(C)
& v1_funct_2(C,B,k1_zfmisc_1(u1_struct_0(A)))
& m1_relset_1(C,B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(D,B) )
=> k6_urysohn1(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k7_urysohn1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_relat_1(k7_urysohn1(A))
& v1_funct_1(k7_urysohn1(A))
& v1_finseq_1(k7_urysohn1(A)) ) ) ).
fof(dt_k8_urysohn1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k3_urysohn1(A)) )
=> m2_subset_1(k8_urysohn1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k9_urysohn1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k9_urysohn1(A,B,C),k1_numbers) ) ).
fof(redefinition_k9_urysohn1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m1_subset_1(C,u1_struct_0(A)) )
=> k9_urysohn1(A,B,C) = k1_funct_1(B,C) ) ).
%------------------------------------------------------------------------------