SET007 Axioms: SET007+506.ax
%------------------------------------------------------------------------------
% File : SET007+506 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Baire Category Theorem
% 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 : waybel12 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 77 ( 3 unt; 0 def)
% Number of atoms : 697 ( 26 equ)
% Maximal formula atoms : 29 ( 9 avg)
% Number of connectives : 708 ( 88 ~; 2 |; 404 &)
% ( 12 <=>; 202 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 56 ( 54 usr; 1 prp; 0-3 aty)
% Number of functors : 30 ( 30 usr; 2 con; 0-4 aty)
% Number of variables : 207 ( 189 !; 18 ?)
% 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_waybel12,axiom,
? [A] :
( l1_struct_0(A)
& v3_struct_0(A) ) ).
fof(fc1_waybel12,axiom,
! [A] :
( ( v3_struct_0(A)
& l1_struct_0(A) )
=> ( v1_xboole_0(u1_struct_0(A))
& v1_finset_1(u1_struct_0(A)) ) ) ).
fof(cc1_waybel12,axiom,
! [A] :
( ( v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_xboole_0(B)
& v1_finset_1(B) ) ) ) ).
fof(cc2_waybel12,axiom,
! [A] :
( v1_finset_1(A)
=> v1_card_4(A) ) ).
fof(rc2_waybel12,axiom,
? [A] :
( v1_xboole_0(A)
& v1_finset_1(A)
& v1_card_4(A) ) ).
fof(rc3_waybel12,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_xboole_0(B)
& v1_finset_1(B)
& v1_card_4(B) ) ) ).
fof(rc4_waybel12,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_card_4(A) ) ).
fof(rc5_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_card_4(B) ) ) ).
fof(rc6_waybel12,axiom,
? [A] :
( ~ v1_finset_1(A)
& v1_card_4(A) ) ).
fof(rc7_waybel12,axiom,
! [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_finset_1(B)
& v1_card_4(B) ) ) ).
fof(cc3_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_waybel_6(B,A)
& v12_waybel_0(B,A) )
=> v2_waybel_0(B,A) ) ) ) ).
fof(cc4_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v12_waybel_0(B,A)
=> v1_waybel_6(B,A) ) ) ) ).
fof(fc2_waybel12,axiom,
! [A,B] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_waybel_3(A)
& v2_lattice3(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_waybel_6(k3_subset_1(u1_struct_0(A),k6_waybel_0(A,B)),A)
& v13_waybel_0(k3_subset_1(u1_struct_0(A),k6_waybel_0(A,B)),A) ) ) ).
fof(rc8_waybel12,axiom,
! [A,B] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A)
& ~ v1_xboole_0(B)
& v2_waybel_0(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( m1_waybel12(C,A,B)
& ~ v1_xboole_0(C) ) ) ).
fof(fc3_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> v2_waybel12(k4_yellow_0(A),A) ) ).
fof(rc9_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& v2_waybel12(B,A) ) ) ).
fof(rc10_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_card_4(B)
& v3_waybel12(B,A) ) ) ).
fof(d1_waybel12,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(B,A)
<=> v3_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ).
fof(d2_waybel12,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_waybel12(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v1_tops_1(C,A) ) ) ) ) ) ).
fof(t1_waybel12,axiom,
$true ).
fof(t2_waybel12,axiom,
! [A,B] :
( ( r1_tarski(k1_card_1(A),k1_card_1(B))
& v1_card_4(B) )
=> v1_card_4(A) ) ).
fof(t3_waybel12,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_4(A) )
=> r2_tarski(k5_numbers,A) ) ).
fof(t4_waybel12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_card_4(A) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,A)
& m2_relset_1(B,k5_numbers,A)
& k2_relat_1(B) = A ) ) ).
fof(t5_waybel12,axiom,
$true ).
fof(t6_waybel12,axiom,
$true ).
fof(t7_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_orders_2(A)
& l1_orders_2(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_yellow_4(A,B,C)
=> r1_tarski(k4_waybel_0(A,B),k4_waybel_0(A,C)) ) ) ) ) ).
fof(t8_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_orders_2(A)
& l1_orders_2(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)))
=> ( r2_yellow_4(A,C,B)
=> r1_tarski(k5_waybel_0(A,B),k5_waybel_0(A,C)) ) ) ) ) ).
fof(t9_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v2_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_yellow_0(A,B)
=> r2_hidden(k2_yellow_0(A,B),B) ) ) ) ).
fof(t10_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v12_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> r2_hidden(k3_yellow_0(A),B) ) ) ).
fof(t11_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> r2_hidden(k3_yellow_0(A),k4_waybel_0(A,B)) ) ) ).
fof(t12_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> r2_hidden(k4_yellow_0(A),B) ) ) ).
fof(t13_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> r2_hidden(k4_yellow_0(A),k5_waybel_0(A,B)) ) ) ).
fof(t14_waybel12,axiom,
! [A] :
( ( v4_orders_2(A)
& v1_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k4_yellow_4(A,B,k1_struct_0(A,k3_yellow_0(A))),k1_struct_0(A,k3_yellow_0(A))) ) ) ).
fof(t15_waybel12,axiom,
! [A] :
( ( v4_orders_2(A)
& v1_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k4_yellow_4(A,B,k1_struct_0(A,k3_yellow_0(A))) = k1_struct_0(A,k3_yellow_0(A)) ) ) ).
fof(t16_waybel12,axiom,
! [A] :
( ( v4_orders_2(A)
& v2_yellow_0(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k2_yellow_4(A,B,k1_struct_0(A,k4_yellow_0(A))),k1_struct_0(A,k4_yellow_0(A))) ) ) ).
fof(t17_waybel12,axiom,
! [A] :
( ( v4_orders_2(A)
& v2_yellow_0(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k2_yellow_4(A,B,k1_struct_0(A,k4_yellow_0(A))) = k1_struct_0(A,k4_yellow_0(A)) ) ) ).
fof(t18_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_yellow_4(A,k1_struct_0(A,k4_yellow_0(A)),B) = B ) ) ).
fof(t19_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k2_yellow_4(A,k1_struct_0(A,k3_yellow_0(A)),B) = B ) ) ).
fof(t20_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(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)
=> ( r3_yellow_4(A,B,C)
& r4_yellow_4(A,C,B) ) ) ) ) ) ).
fof(t21_waybel12,axiom,
! [A] :
( ( v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_orders_2(A,C,D)
=> r2_yellow_4(A,k4_yellow_4(A,k1_struct_0(A,C),B),k4_yellow_4(A,k1_struct_0(A,D),B)) ) ) ) ) ) ).
fof(t22_waybel12,axiom,
! [A] :
( ( v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_orders_2(A,C,D)
=> r1_yellow_4(A,k2_yellow_4(A,k1_struct_0(A,C),B),k2_yellow_4(A,k1_struct_0(A,D),B)) ) ) ) ) ) ).
fof(t23_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_yellow_4(A,D,C)
& v13_waybel_0(B,A)
& r1_tarski(D,B) )
=> r1_tarski(C,B) ) ) ) ) ) ).
fof(t24_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_yellow_4(A,C,D)
& v12_waybel_0(B,A)
& r1_tarski(D,B) )
=> r1_tarski(C,B) ) ) ) ) ) ).
fof(t25_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v2_waybel_0(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k4_yellow_4(A,B,B) = B ) ) ).
fof(t26_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_waybel_0(B,A)
& v12_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k2_yellow_4(A,B,B) = B ) ) ).
fof(t30_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_waybel_6(B,A)
& v12_waybel_0(B,A) )
=> v2_waybel_0(B,A) ) ) ) ).
fof(t31_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& ~ r3_orders_2(A,C,D)
& ~ r3_orders_2(A,D,C) ) ) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> r2_hidden(k2_yellow_0(A,C),C) ) ) ) ) ).
fof(t32_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& ~ r3_orders_2(A,C,D)
& ~ r3_orders_2(A,D,C) ) ) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> r2_hidden(k1_yellow_0(A,C),C) ) ) ) ) ).
fof(d3_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_waybel_0(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_waybel12(C,A,B)
<=> B = k5_waybel_0(A,k13_waybel_0(A,C)) ) ) ) ) ).
fof(t33_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r4_yellow_4(A,C,B)
=> r4_yellow_4(A,k13_waybel_0(A,C),k13_waybel_0(A,B)) ) ) ) ) ).
fof(t34_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_waybel_0(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_waybel12(C,A,B)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ( r4_yellow_4(A,D,C)
& r4_yellow_4(A,B,D) )
=> m1_waybel12(D,A,B) ) ) ) ) ) ).
fof(t37_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v3_waybel_3(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_waybel_6(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v2_waybel_0(C,A)
& v13_waybel_0(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r1_tarski(k4_yellow_4(A,B,C),B)
& r2_hidden(D,B)
& ? [E] :
( ~ v1_xboole_0(E)
& m1_waybel12(E,A,C)
& v1_card_4(E) )
& ! [E] :
( ( ~ v1_xboole_0(E)
& v1_waybel_6(E,A)
& v2_waybel_0(E,A)
& v13_waybel_0(E,A)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(E,B)
& r2_hidden(D,E)
& r1_tarski(C,E) ) ) ) ) ) ) ) ).
fof(t38_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v3_waybel_3(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_waybel_6(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r1_tarski(k4_yellow_4(A,B,C),B)
& r2_hidden(D,B)
& ! [E] :
( ( ~ v1_xboole_0(E)
& v1_waybel_6(E,A)
& v2_waybel_0(E,A)
& v13_waybel_0(E,A)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(k4_yellow_4(A,k1_struct_0(A,D),C),E)
& r1_tarski(E,B)
& r2_hidden(D,E) ) ) ) ) ) ) ) ).
fof(t39_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v3_waybel_3(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_waybel_6(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r1_tarski(k4_yellow_4(A,B,C),B)
& r2_hidden(E,B)
& ~ r2_hidden(D,B)
& ! [F] :
( ( v2_waybel_6(F,A)
& m1_subset_1(F,u1_struct_0(A)) )
=> ~ ( r3_orders_2(A,D,F)
& ~ r2_hidden(F,k5_waybel_0(A,k4_yellow_4(A,k1_struct_0(A,E),C))) ) ) ) ) ) ) ) ) ).
fof(t40_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v3_waybel_3(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( ~ r3_orders_2(A,E,B)
& r2_hidden(D,C)
& r3_orders_2(A,k12_lattice3(A,E,D),B) ) ) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( ~ r3_orders_2(A,D,B)
& ! [E] :
( ( v2_waybel_6(E,A)
& m1_subset_1(E,u1_struct_0(A)) )
=> ~ ( r3_orders_2(A,B,E)
& ~ r2_hidden(E,k5_waybel_0(A,k4_yellow_4(A,k1_struct_0(A,D),C))) ) ) ) ) ) ) ) ) ).
fof(d4_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v2_waybel12(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( C != k3_yellow_0(A)
& k11_lattice3(A,B,C) = k3_yellow_0(A) ) ) ) ) ) ).
fof(t41_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_yellow_0(A)
& v2_lattice3(A)
& ~ v3_realset2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( v2_waybel12(B,A)
& B = k3_yellow_0(A) ) ) ) ).
fof(d5_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_waybel12(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> v2_waybel12(C,A) ) ) ) ) ) ).
fof(t42_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> v3_waybel12(k1_struct_0(A,k4_yellow_0(A)),A) ) ).
fof(t43_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v3_waybel_3(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_card_4(B)
& v3_waybel12(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( C != k3_yellow_0(A)
& ! [D] :
( ( v2_waybel_6(D,A)
& m1_subset_1(D,u1_struct_0(A)) )
=> ~ ( D != k4_yellow_0(A)
& ~ r2_hidden(D,k5_waybel_0(A,k4_yellow_4(A,k1_struct_0(A,C),B))) ) ) ) ) ) ) ).
fof(t44_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( B = C
& v2_yellow_8(k3_subset_1(u1_struct_0(A),C),A) )
=> v2_waybel_6(B,k2_yellow_1(u1_pre_topc(A))) ) ) ) ) ).
fof(t45_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = C
=> ( B = k4_yellow_0(k2_yellow_1(u1_pre_topc(A)))
| ( v2_waybel_6(B,k2_yellow_1(u1_pre_topc(A)))
<=> v2_yellow_8(k3_subset_1(u1_struct_0(A),C),A) ) ) ) ) ) ) ).
fof(t46_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = C
=> ( v2_waybel12(B,k2_yellow_1(u1_pre_topc(A)))
<=> v1_tops_3(C,A) ) ) ) ) ) ).
fof(t47_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v6_waybel_3(A)
=> ! [B] :
( ( v1_card_4(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> ( ( v1_waybel12(B,A)
& v1_tops_2(B,A) )
=> ( v1_xboole_0(B)
| ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v3_pre_topc(C,A)
& ! [D] :
( ( v2_yellow_8(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(E,B)
& r1_xboole_0(k3_finsub_1(k1_zfmisc_1(u1_struct_0(A)),D,C),E) ) ) ) ) ) ) ) ) ) ).
fof(d6_waybel12,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_pre_topc(C,A)
& v1_tops_1(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(t48_waybel12,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v3_yellow_8(A)
& v6_waybel_3(A) )
=> v1_yellow_8(A) ) ) ).
fof(dt_m1_waybel12,axiom,
! [A,B] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A)
& ~ v1_xboole_0(B)
& v2_waybel_0(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_waybel12(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(existence_m1_waybel12,axiom,
! [A,B] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A)
& ~ v1_xboole_0(B)
& v2_waybel_0(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] : m1_waybel12(C,A,B) ) ).
fof(t27_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ v1_xboole_0(a_2_0_waybel12(A,B)) ) ) ).
fof(t28_waybel12,axiom,
! [A] :
( ( v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_waybel_0(a_2_1_waybel12(A,B),A)
& m1_subset_1(a_2_1_waybel12(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(t29_waybel12,axiom,
! [A] :
( ( v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v13_waybel_0(a_2_2_waybel12(A,B),A)
& m1_subset_1(a_2_2_waybel12(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(t35_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( ( k8_yellow_2(k5_numbers,A,C) = B
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k7_yellow_2(k5_numbers,A,D,E) = k2_yellow_0(A,a_3_0_waybel12(A,C,E)) ) )
=> r4_yellow_4(A,k8_yellow_2(k5_numbers,A,D),B) ) ) ) ) ) ).
fof(t36_waybel12,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_waybel_0(B,A)
& v13_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_waybel12(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(A))
& m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
=> ( ( k8_yellow_2(k5_numbers,A,D) = C
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k7_yellow_2(k5_numbers,A,E,F) = k2_yellow_0(A,a_3_0_waybel12(A,D,F)) ) )
=> m1_waybel12(k8_yellow_2(k5_numbers,A,E),A,B) ) ) ) ) ) ) ).
fof(fraenkel_a_2_0_waybel12,axiom,
! [A,B,C] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v2_yellow_0(B)
& v2_lattice3(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_waybel12(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r1_tarski(k4_yellow_4(B,C,k1_struct_0(B,D)),C) ) ) ) ).
fof(fraenkel_a_2_1_waybel12,axiom,
! [A,B,C] :
( ( v3_orders_2(B)
& v4_orders_2(B)
& v2_lattice3(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_1_waybel12(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r1_tarski(k4_yellow_4(B,C,k1_struct_0(B,D)),C) ) ) ) ).
fof(fraenkel_a_2_2_waybel12,axiom,
! [A,B,C] :
( ( v3_orders_2(B)
& v4_orders_2(B)
& v2_lattice3(B)
& l1_orders_2(B)
& v13_waybel_0(C,B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_2_waybel12(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r1_tarski(k4_yellow_4(B,C,k1_struct_0(B,D)),C) ) ) ) ).
fof(fraenkel_a_3_0_waybel12,axiom,
! [A,B,C,D] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v2_lattice3(B)
& l1_orders_2(B)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(B))
& m2_relset_1(C,k5_numbers,u1_struct_0(B))
& m2_subset_1(D,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_3_0_waybel12(B,C,D))
<=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& A = k7_yellow_2(k5_numbers,B,C,E)
& r1_xreal_0(E,D) ) ) ) ).
%------------------------------------------------------------------------------