SET007 Axioms: SET007+489.ax
%------------------------------------------------------------------------------
% File : SET007+489 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The ``Way-Below'' Relation
% 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 : waybel_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 86 ( 1 unt; 0 def)
% Number of atoms : 809 ( 18 equ)
% Maximal formula atoms : 23 ( 9 avg)
% Number of connectives : 839 ( 116 ~; 2 |; 464 &)
% ( 20 <=>; 237 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 57 ( 56 usr; 0 prp; 1-3 aty)
% Number of functors : 30 ( 30 usr; 0 con; 1-4 aty)
% Number of variables : 247 ( 238 !; 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(fc1_waybel_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ~ v1_xboole_0(k1_waybel_3(A,B)) ) ).
fof(fc2_waybel_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> v12_waybel_0(k1_waybel_3(A,B),A) ) ).
fof(fc3_waybel_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> v13_waybel_0(k2_waybel_3(A,B),A) ) ).
fof(fc4_waybel_3,axiom,
! [A,B] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_waybel_0(k1_waybel_3(A,B),A)
& v12_waybel_0(k1_waybel_3(A,B),A) ) ) ).
fof(fc5_waybel_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v25_waybel_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v1_xboole_0(k1_waybel_3(A,B))
& v1_waybel_0(k1_waybel_3(A,B),A)
& v12_waybel_0(k1_waybel_3(A,B),A) ) ) ).
fof(cc1_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v16_waybel_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_waybel_0(B,A)
& v2_waybel_0(B,A) ) ) ) ).
fof(cc2_waybel_3,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v16_waybel_0(A)
& v24_waybel_0(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v1_yellow_0(A)
& v2_yellow_0(A)
& v3_yellow_0(A)
& v16_waybel_0(A)
& v24_waybel_0(A)
& v25_waybel_0(A) ) ) ) ).
fof(rc1_waybel_3,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v1_yellow_0(A)
& v2_yellow_0(A)
& v3_yellow_0(A)
& v16_waybel_0(A)
& v24_waybel_0(A)
& v25_waybel_0(A) ) ).
fof(cc3_waybel_3,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A)
& v2_orders_2(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v2_waybel_3(A) ) ) ) ).
fof(cc4_waybel_3,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_waybel_3(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v24_waybel_0(A)
& v2_waybel_3(A) ) ) ) ).
fof(cc5_waybel_3,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v1_yellow_0(A)
& v24_waybel_0(A)
& v2_waybel_3(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v1_yellow_0(A)
& v3_waybel_3(A) ) ) ) ).
fof(rc2_waybel_3,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_orders_2(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v1_yellow_0(A)
& v2_yellow_0(A)
& v3_yellow_0(A)
& v24_waybel_0(A)
& v25_waybel_0(A)
& v2_waybel_3(A)
& v3_waybel_3(A) ) ).
fof(fc6_waybel_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_waybel_3(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v1_xboole_0(k1_waybel_3(A,B))
& v1_waybel_0(k1_waybel_3(A,B),A) ) ) ).
fof(cc6_waybel_3,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& v16_waybel_0(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v16_waybel_0(A)
& v2_waybel_3(A) ) ) ) ).
fof(rc3_waybel_3,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_yellow_1(B)
& v2_pralg_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B) ) ).
fof(fc7_waybel_3,axiom,
! [A,B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k5_yellow_1(A,B))
& v1_orders_2(k5_yellow_1(A,B)) ) ) ).
fof(fc8_waybel_3,axiom,
! [A,B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k5_yellow_1(A,B))
& v1_orders_2(k5_yellow_1(A,B))
& v1_monoid_0(k5_yellow_1(A,B)) ) ) ).
fof(fc9_waybel_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k5_yellow_1(A,B))
& v1_orders_2(k5_yellow_1(A,B))
& v2_orders_2(k5_yellow_1(A,B))
& v1_monoid_0(k5_yellow_1(A,B)) ) ) ).
fof(cc7_waybel_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_compts_1(A)
& v3_compts_1(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v4_compts_1(A)
& v5_compts_1(A)
& v6_waybel_3(A) ) ) ) ).
fof(rc4_waybel_3,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_compts_1(A)
& v3_compts_1(A)
& v4_compts_1(A)
& v5_compts_1(A)
& v6_waybel_3(A) ) ).
fof(d1_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_waybel_3(A,B,C)
<=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_waybel_0(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r3_orders_2(A,C,k1_yellow_0(A,D))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,D)
& r3_orders_2(A,B,E) ) ) ) ) ) ) ) ) ).
fof(d2_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_waybel_3(B,A)
<=> r1_waybel_3(A,B,B) ) ) ) ).
fof(t1_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_waybel_3(A,B,C)
=> r3_orders_2(A,B,C) ) ) ) ) ).
fof(t2_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,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,B,C)
& r1_waybel_3(A,C,D)
& r3_orders_2(A,D,E) )
=> r1_waybel_3(A,B,E) ) ) ) ) ) ) ).
fof(t3_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ( ( v1_lattice3(A)
| v25_waybel_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r1_waybel_3(A,B,D)
& r1_waybel_3(A,C,D) )
=> ( r1_yellow_0(A,k2_struct_0(A,B,C))
& r1_waybel_3(A,k10_lattice3(A,B,C),D) ) ) ) ) ) ) ) ).
fof(t4_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_waybel_3(A,k3_yellow_0(A),B) ) ) ).
fof(t5_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r1_waybel_3(A,B,C)
& r1_waybel_3(A,C,D) )
=> r1_waybel_3(A,B,D) ) ) ) ) ) ).
fof(t6_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( r1_waybel_3(A,B,C)
& r1_waybel_3(A,C,B) )
=> B = C ) ) ) ) ).
fof(t7_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(B,k1_waybel_3(A,C))
<=> r1_waybel_3(A,B,C) ) ) ) ) ).
fof(t8_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(B,k2_waybel_3(A,C))
<=> r1_waybel_3(A,C,B) ) ) ) ) ).
fof(t9_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_lattice3(A,k1_waybel_3(A,B),B) ) ) ).
fof(t10_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_lattice3(A,k2_waybel_3(A,B),B) ) ) ).
fof(t11_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r1_tarski(k1_waybel_3(A,B),k6_waybel_0(A,B))
& r1_tarski(k2_waybel_3(A,B),k7_waybel_0(A,B)) ) ) ) ).
fof(t12_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_orders_2(A,B,C)
=> ( r1_tarski(k1_waybel_3(A,B),k1_waybel_3(A,C))
& r1_tarski(k2_waybel_3(A,C),k2_waybel_3(A,B)) ) ) ) ) ) ).
fof(t13_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v16_waybel_0(A)
& v24_waybel_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_orders_2(A,B,C)
=> r1_waybel_3(A,B,C) ) ) ) ) ).
fof(t14_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_waybel_3(A,B,C)
=> ( v1_waybel_3(B,A)
| r2_orders_2(A,B,C) ) ) ) ) ) ).
fof(t15_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& l1_orders_2(A) )
=> v1_waybel_3(k3_yellow_0(A),A) ) ).
fof(t16_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v24_waybel_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> r2_hidden(k1_yellow_0(A,B),B) ) ) ).
fof(t17_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v24_waybel_0(A)
& l1_orders_2(A) )
=> ( v6_group_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v1_waybel_3(B,A) ) ) ) ).
fof(t18_waybel_3,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_waybel_3(A,B,C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r3_orders_2(A,C,k1_yellow_0(A,D))
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(E,D)
& r3_orders_2(A,B,k1_yellow_0(A,E)) ) ) ) ) ) ) ) ) ).
fof(t19_waybel_3,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r3_orders_2(A,C,k1_yellow_0(A,D))
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(E,D)
& r3_orders_2(A,B,k1_yellow_0(A,E)) ) ) ) )
=> r1_waybel_3(A,B,C) ) ) ) ) ).
fof(t20_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_waybel_3(A,B,C)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_waybel_0(D,A)
& v12_waybel_0(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r3_orders_2(A,C,k1_yellow_0(A,D))
=> r2_hidden(B,D) ) ) ) ) ) ) ).
fof(t21_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v24_waybel_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ! [D] :
( ( ~ v1_xboole_0(D)
& v1_waybel_0(D,A)
& v12_waybel_0(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r3_orders_2(A,C,k1_yellow_0(A,D))
=> r2_hidden(B,D) ) )
=> r1_waybel_3(A,B,C) ) ) ) ) ).
fof(t22_waybel_3,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v1_yellow_0(A)
& l1_orders_2(A) )
=> ( v2_waybel_2(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_waybel_3(A,B,C)
<=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_waybel_0(D,A)
& v12_waybel_0(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( C = k1_yellow_0(A,D)
=> r2_hidden(B,D) ) ) ) ) ) ) ) ).
fof(t23_waybel_3,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v1_waybel_3(B,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))
& r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,B)
& r2_orders_2(A,C,D) ) ) ) ) ) ) ).
fof(d5_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A) )
=> ( v2_waybel_3(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> B = k1_yellow_0(A,k1_waybel_3(A,B)) ) ) ) ).
fof(d6_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A) )
=> ( v3_waybel_3(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ~ v1_xboole_0(k1_waybel_3(A,B))
& v1_waybel_0(k1_waybel_3(A,B),A) ) )
& v24_waybel_0(A)
& v2_waybel_3(A) ) ) ) ).
fof(t24_waybel_3,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& v24_waybel_0(A)
& l1_orders_2(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ~ v1_xboole_0(k1_waybel_3(A,B))
& v1_waybel_0(k1_waybel_3(A,B),A) ) )
=> ( v2_waybel_3(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( ~ r3_orders_2(A,B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r1_waybel_3(A,D,B)
& ~ r3_orders_2(A,D,C) ) ) ) ) ) ) ) ) ).
fof(t25_waybel_3,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_orders_2(A,B,C)
<=> r1_tarski(k1_waybel_3(A,B),k1_waybel_3(A,C)) ) ) ) ) ).
fof(t26_waybel_3,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v1_waybel_3(B,A) )
=> v2_waybel_3(A) ) ) ).
fof(d7_waybel_3,axiom,
! [A] :
( v1_relat_1(A)
=> ( v4_waybel_3(A)
<=> ! [B] :
( l1_struct_0(B)
=> ~ ( r2_hidden(B,k2_relat_1(A))
& v3_struct_0(B) ) ) ) ) ).
fof(d8_waybel_3,axiom,
! [A] :
( v1_relat_1(A)
=> ( v5_waybel_3(A)
<=> ! [B] :
( l1_orders_2(B)
=> ( r2_hidden(B,k2_relat_1(A))
=> v2_orders_2(B) ) ) ) ) ).
fof(t27_waybel_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
<=> ( k1_relat_1(C) = A
& ! [D] :
( m1_subset_1(D,A)
=> m1_subset_1(k1_funct_1(C,D),u1_struct_0(k3_waybel_3(A,B,D))) ) ) ) ) ) ) ).
fof(t28_waybel_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_yellow_1(A,B)))
=> ( r1_orders_2(k5_yellow_1(A,B),C,D)
<=> ! [E] :
( m1_subset_1(E,A)
=> r1_orders_2(k3_waybel_3(A,B,E),k4_waybel_3(A,B,C,E),k4_waybel_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t29_waybel_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v3_orders_2(k3_waybel_3(A,B,C)) )
=> v3_orders_2(k5_yellow_1(A,B)) ) ) ) ).
fof(t30_waybel_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> v4_orders_2(k3_waybel_3(A,B,C)) )
=> v4_orders_2(k5_yellow_1(A,B)) ) ) ) ).
fof(t31_waybel_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v2_orders_2(k6_waybel_3(A,B,C))
& v3_orders_2(k6_waybel_3(A,B,C))
& v4_orders_2(k6_waybel_3(A,B,C))
& v1_lattice3(k6_waybel_3(A,B,C))
& v2_lattice3(k6_waybel_3(A,B,C))
& v3_lattice3(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) )
=> ( v2_orders_2(k5_yellow_1(A,B))
& v3_orders_2(k5_yellow_1(A,B))
& v4_orders_2(k5_yellow_1(A,B))
& v1_lattice3(k5_yellow_1(A,B))
& v2_lattice3(k5_yellow_1(A,B))
& v3_lattice3(k5_yellow_1(A,B))
& l1_orders_2(k5_yellow_1(A,B)) ) ) ) ) ).
fof(t32_waybel_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v2_orders_2(k6_waybel_3(A,B,C))
& v3_orders_2(k6_waybel_3(A,B,C))
& v4_orders_2(k6_waybel_3(A,B,C))
& v1_lattice3(k6_waybel_3(A,B,C))
& v2_lattice3(k6_waybel_3(A,B,C))
& v3_lattice3(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_yellow_1(A,B))))
=> ! [D] :
( m1_subset_1(D,A)
=> k7_waybel_3(A,B,k1_yellow_0(k5_yellow_1(A,B),C),D) = k1_yellow_0(k3_waybel_3(A,B,D),k5_waybel_3(A,B,D,C)) ) ) ) ) ) ).
fof(t33_waybel_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v2_orders_2(k6_waybel_3(A,B,C))
& v3_orders_2(k6_waybel_3(A,B,C))
& v4_orders_2(k6_waybel_3(A,B,C))
& v1_lattice3(k6_waybel_3(A,B,C))
& v2_lattice3(k6_waybel_3(A,B,C))
& v3_lattice3(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_yellow_1(A,B)))
=> ( r1_waybel_3(k5_yellow_1(A,B),C,D)
<=> ( ! [E] :
( m1_subset_1(E,A)
=> r1_waybel_3(k6_waybel_3(A,B,E),k7_waybel_3(A,B,C,E),k7_waybel_3(A,B,D,E)) )
& ? [E] :
( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(A))
& ! [F] :
( m1_subset_1(F,A)
=> ( ~ r2_hidden(F,E)
=> k7_waybel_3(A,B,C,F) = k3_yellow_0(k6_waybel_3(A,B,F)) ) ) ) ) ) ) ) ) ) ) ).
fof(t34_waybel_3,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,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ( r1_waybel_3(k2_yellow_1(u1_pre_topc(A)),B,C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_tops_2(D,A)
& r1_tarski(C,k5_setfam_1(u1_struct_0(A),D))
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(D)) )
=> ~ r1_tarski(B,k3_tarski(E)) ) ) ) ) ) ) ) ).
fof(t35_waybel_3,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,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_tops_2(D,A)
& r1_tarski(C,k5_setfam_1(u1_struct_0(A),D))
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(D)) )
=> ~ r1_tarski(B,k3_tarski(E)) ) ) )
=> r1_waybel_3(k2_yellow_1(u1_pre_topc(A)),B,C) ) ) ) ) ).
fof(t36_waybel_3,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
=> ( v1_waybel_3(B,k2_yellow_1(u1_pre_topc(A)))
<=> v6_compts_1(C,A) ) ) ) ) ) ).
fof(t37_waybel_3,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))))
=> ( B = u1_struct_0(A)
=> ( v1_waybel_3(B,k2_yellow_1(u1_pre_topc(A)))
<=> v2_compts_1(A) ) ) ) ) ).
fof(d9_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(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,C)
& v3_pre_topc(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,k1_tops_1(A,D))
& r1_tarski(D,C)
& v6_compts_1(D,A) ) ) ) ) ) ) ) ).
fof(t38_waybel_3,axiom,
! [A] : v3_compts_1(k2_pcomps_1(k1_tarski(A))) ).
fof(t39_waybel_3,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,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ( ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(B,D)
& r1_tarski(D,C)
& v6_compts_1(D,A) )
=> r1_waybel_3(k2_yellow_1(u1_pre_topc(A)),B,C) ) ) ) ) ).
fof(t40_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v6_waybel_3(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ~ ( r1_waybel_3(k2_yellow_1(u1_pre_topc(A)),B,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(B,D)
& r1_tarski(D,C)
& v6_compts_1(D,A) ) ) ) ) ) ) ) ).
fof(t41_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v6_waybel_3(A)
& v3_compts_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k2_yellow_1(u1_pre_topc(A))))
=> ~ ( r1_waybel_3(k2_yellow_1(u1_pre_topc(A)),B,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( D = B
& r1_tarski(k6_pre_topc(A,D),C)
& v6_compts_1(k6_pre_topc(A,D),A) ) ) ) ) ) ) ) ).
fof(t42_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v4_compts_1(A)
& v3_waybel_3(k2_yellow_1(u1_pre_topc(A))) )
=> v6_waybel_3(A) ) ) ).
fof(t43_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v6_waybel_3(A)
=> v3_waybel_3(k2_yellow_1(u1_pre_topc(A))) ) ) ).
fof(dt_k1_waybel_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k1_waybel_3(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_waybel_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k2_waybel_3(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k3_waybel_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> ( ~ v3_struct_0(k3_waybel_3(A,B,C))
& l1_orders_2(k3_waybel_3(A,B,C)) ) ) ).
fof(redefinition_k3_waybel_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> k3_waybel_3(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k4_waybel_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
& m1_subset_1(D,A) )
=> m1_subset_1(k4_waybel_3(A,B,C,D),u1_struct_0(k3_waybel_3(A,B,D))) ) ).
fof(redefinition_k4_waybel_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
& m1_subset_1(D,A) )
=> k4_waybel_3(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k5_waybel_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_yellow_1(A,B)))) )
=> m1_subset_1(k5_waybel_3(A,B,C,D),k1_zfmisc_1(u1_struct_0(k3_waybel_3(A,B,C)))) ) ).
fof(redefinition_k5_waybel_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_yellow_1(A,B)))) )
=> k5_waybel_3(A,B,C,D) = k5_card_3(C,D) ) ).
fof(dt_k6_waybel_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> ( ~ v3_struct_0(k6_waybel_3(A,B,C))
& v2_orders_2(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) ) ).
fof(redefinition_k6_waybel_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> k6_waybel_3(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k7_waybel_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
& m1_subset_1(D,A) )
=> m1_subset_1(k7_waybel_3(A,B,C,D),u1_struct_0(k6_waybel_3(A,B,D))) ) ).
fof(redefinition_k7_waybel_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A)
& m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
& m1_subset_1(D,A) )
=> k7_waybel_3(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(d3_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_waybel_3(A,B) = a_2_0_waybel_3(A,B) ) ) ).
fof(d4_waybel_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_waybel_3(A,B) = a_2_1_waybel_3(A,B) ) ) ).
fof(fraenkel_a_2_0_waybel_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& l1_orders_2(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_waybel_3(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r1_waybel_3(B,D,C) ) ) ) ).
fof(fraenkel_a_2_1_waybel_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& l1_orders_2(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_1_waybel_3(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r1_waybel_3(B,C,D) ) ) ) ).
%------------------------------------------------------------------------------