SET007 Axioms: SET007+502.ax
%------------------------------------------------------------------------------
% File : SET007+502 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Baire Spaces, Sober 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 : yellow_8 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 9 unt; 0 def)
% Number of atoms : 333 ( 21 equ)
% Maximal formula atoms : 13 ( 5 avg)
% Number of connectives : 330 ( 57 ~; 1 |; 130 &)
% ( 15 <=>; 127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-3 aty)
% Number of functors : 20 ( 20 usr; 0 con; 1-3 aty)
% Number of variables : 131 ( 122 !; 9 ?)
% 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(cc1_yellow_8,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_yellow_8(B,A)
=> ~ v1_xboole_0(B) ) ) ) ).
fof(rc1_yellow_8,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_yellow_8(B,A) ) ) ).
fof(cc2_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_yellow_8(B,A)
=> v1_realset1(B) ) ) ) ).
fof(cc3_yellow_8,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_compts_1(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_yellow_8(A) ) ) ) ).
fof(rc2_yellow_8,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_yellow_8(A) ) ).
fof(cc4_yellow_8,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_yellow_8(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_t_0topsp(A) ) ) ) ).
fof(fc1_yellow_8,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k1_yellow_8(A))
& v1_pre_topc(k1_yellow_8(A)) ) ) ).
fof(fc2_yellow_8,axiom,
! [A] :
( v1_pre_topc(k1_yellow_8(A))
& v2_pre_topc(k1_yellow_8(A)) ) ).
fof(fc3_yellow_8,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k1_yellow_8(A))
& v1_pre_topc(k1_yellow_8(A))
& v2_pre_topc(k1_yellow_8(A))
& v1_urysohn1(k1_yellow_8(A)) ) ) ).
fof(fc4_yellow_8,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ( ~ v3_struct_0(k1_yellow_8(A))
& v1_pre_topc(k1_yellow_8(A))
& v2_pre_topc(k1_yellow_8(A))
& v1_urysohn1(k1_yellow_8(A))
& ~ v3_yellow_8(k1_yellow_8(A)) ) ) ).
fof(rc3_yellow_8,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_urysohn1(A)
& ~ v3_yellow_8(A) ) ).
fof(t1_yellow_8,axiom,
! [A,B,C] :
( ( r2_hidden(B,k5_finsub_1(A))
& r1_tarski(C,B) )
=> r2_hidden(C,k5_finsub_1(A)) ) ).
fof(t2_yellow_8,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r1_tarski(B,k5_finsub_1(A))
=> r2_hidden(k6_setfam_1(A,B),k5_finsub_1(A)) ) ) ).
fof(d1_yellow_8,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_realset1(A)
<=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> B = C ) ) ) ) ).
fof(t3_yellow_8,axiom,
$true ).
fof(t4_yellow_8,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> r2_wellord2(B,k7_setfam_1(A,B)) ) ).
fof(t5_yellow_8,axiom,
! [A,B] :
( ( r2_wellord2(A,B)
& v1_card_4(A) )
=> v1_card_4(B) ) ).
fof(t6_yellow_8,axiom,
$true ).
fof(t7_yellow_8,axiom,
$true ).
fof(t8_yellow_8,axiom,
$true ).
fof(t9_yellow_8,axiom,
$true ).
fof(t10_yellow_8,axiom,
$true ).
fof(t11_yellow_8,axiom,
$true ).
fof(t12_yellow_8,axiom,
$true ).
fof(t13_yellow_8,axiom,
$true ).
fof(t14_yellow_8,axiom,
! [A] :
( l1_struct_0(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)))
=> ( r2_hidden(k3_subset_1(u1_struct_0(A),C),k7_setfam_1(u1_struct_0(A),B))
<=> r2_hidden(C,B) ) ) ) ) ).
fof(t15_yellow_8,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> k8_setfam_1(u1_struct_0(A),k7_setfam_1(u1_struct_0(A),B)) = k3_subset_1(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) ).
fof(t16_yellow_8,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> k5_setfam_1(u1_struct_0(A),k7_setfam_1(u1_struct_0(A),B)) = k3_subset_1(u1_struct_0(A),k8_setfam_1(u1_struct_0(A),B)) ) ) ).
fof(t17_yellow_8,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(C,B)
& v4_pre_topc(B,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_tarski(C,D)
& v4_pre_topc(D,A) )
=> r1_tarski(B,D) ) ) )
=> B = k6_pre_topc(A,C) ) ) ) ) ).
fof(t19_yellow_8,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_cantor_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v3_pre_topc(C,A) ) ) ) ) ).
fof(d2_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( m1_yellow_8(C,A,B)
<=> ( r1_tarski(C,u1_pre_topc(A))
& r2_hidden(B,k8_setfam_1(u1_struct_0(A),C))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& r2_hidden(B,D)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,C)
& r1_tarski(E,D) ) ) ) ) ) ) ) ) ) ).
fof(t21_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_yellow_8(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(D,C)
=> ( v3_pre_topc(D,A)
& r2_hidden(B,D) ) ) ) ) ) ) ).
fof(t22_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_yellow_8(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,D)
& v3_pre_topc(D,A)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,C)
& r1_tarski(E,D) ) ) ) ) ) ) ) ).
fof(t23_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( r1_tarski(B,u1_pre_topc(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_yellow_8(D,A,C)
& r1_tarski(D,B) ) ) )
=> m1_cantor_1(B,A) ) ) ) ).
fof(d3_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_yellow_8(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_card_4(B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v1_tops_3(C,A) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( C = k8_setfam_1(u1_struct_0(A),B)
& v1_tops_1(C,A) ) ) ) ) ) ) ).
fof(t24_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_yellow_8(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( v1_card_4(B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v3_tops_1(C,A) ) ) )
=> v2_tops_1(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ) ).
fof(d4_yellow_8,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_yellow_8(B,A)
<=> ( ~ v1_xboole_0(B)
& v4_pre_topc(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(C,A)
& v4_pre_topc(D,A)
& B = k1_finsub_1(k1_zfmisc_1(u1_struct_0(A)),C,D)
& C != B
& D != B ) ) ) ) ) ) ) ).
fof(d5_yellow_8,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_yellow_8(A,B,C)
<=> ( r2_hidden(C,B)
& r1_tarski(B,k6_pre_topc(A,k1_struct_0(A,C))) ) ) ) ) ) ).
fof(t25_yellow_8,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))
=> ( r1_yellow_8(A,B,C)
=> B = k6_pre_topc(A,k1_struct_0(A,C)) ) ) ) ) ) ).
fof(t26_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v2_yellow_8(k6_pre_topc(A,k1_struct_0(A,B)),A) ) ) ).
fof(d6_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_yellow_8(A)
<=> ! [B] :
( ( v2_yellow_8(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r1_yellow_8(A,B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_yellow_8(A,B,D)
=> C = D ) ) ) ) ) ) ).
fof(t27_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_yellow_8(A,k6_pre_topc(A,k1_struct_0(A,B)),B) ) ) ).
fof(t28_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_yellow_8(A,k1_struct_0(A,B),B) ) ) ).
fof(t29_yellow_8,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)))
=> ( ( v3_pre_topc(B,A)
& v4_pre_topc(C,A) )
=> v4_pre_topc(k2_finsub_1(k1_zfmisc_1(u1_struct_0(A)),C,B),A) ) ) ) ) ).
fof(t30_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_yellow_8(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v1_realset1(B) ) ) ).
fof(t31_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_compts_1(A)
& l1_pre_topc(A) )
=> v3_yellow_8(A) ) ).
fof(t32_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v2_t_0topsp(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k6_pre_topc(A,k1_struct_0(A,B)) = k6_pre_topc(A,k1_struct_0(A,C))
=> B = C ) ) ) ) ) ).
fof(t33_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_yellow_8(A)
& l1_pre_topc(A) )
=> v2_t_0topsp(A) ) ).
fof(d7_yellow_8,axiom,
! [A,B] :
( ( v1_pre_topc(B)
& l1_pre_topc(B) )
=> ( B = k1_yellow_8(A)
<=> ( u1_struct_0(B) = A
& k7_setfam_1(u1_struct_0(B),u1_pre_topc(B)) = k2_xboole_0(k1_tarski(A),k5_finsub_1(A)) ) ) ) ).
fof(t34_yellow_8,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k1_yellow_8(A))))
=> ( v4_pre_topc(B,k1_yellow_8(A))
<=> ( B = A
| v1_finset_1(B) ) ) ) ) ).
fof(t35_yellow_8,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))
=> k6_pre_topc(A,k1_struct_0(A,B)) = k1_struct_0(A,B) ) ) ) ).
fof(t36_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v4_compts_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,k1_tops_1(A,C))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_pre_topc(D,A)
& r1_tarski(D,C)
& r2_hidden(B,k1_tops_1(A,D)) ) ) ) ) ) ) ) ).
fof(t37_yellow_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v4_compts_1(A)
=> ( v6_waybel_3(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(B,k1_tops_1(A,C))
& v6_compts_1(C,A) ) ) ) ) ) ).
fof(dt_m1_yellow_8,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_yellow_8(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ).
fof(existence_m1_yellow_8,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ? [C] : m1_yellow_8(C,A,B) ) ).
fof(dt_k1_yellow_8,axiom,
! [A] :
( v1_pre_topc(k1_yellow_8(A))
& l1_pre_topc(k1_yellow_8(A)) ) ).
fof(t18_yellow_8,axiom,
! [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 = k3_tarski(a_3_0_yellow_8(A,B,C)) ) ) ) ) ).
fof(t20_yellow_8,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,k1_zfmisc_1(u1_struct_0(A)))
=> k1_tops_1(A,C) = k3_tarski(a_3_1_yellow_8(A,B,C)) ) ) ) ).
fof(fraenkel_a_3_0_yellow_8,axiom,
! [A,B,C,D] :
( ( l1_pre_topc(B)
& m1_cantor_1(C,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_0_yellow_8(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& A = E
& r2_hidden(E,C)
& r1_tarski(E,D) ) ) ) ).
fof(fraenkel_a_3_1_yellow_8,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,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_1_yellow_8(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& A = E
& r2_hidden(E,C)
& r1_tarski(E,D) ) ) ) ).
%------------------------------------------------------------------------------