SET007 Axioms: SET007+287.ax
%------------------------------------------------------------------------------
% File : SET007+287 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Elementary Variants of Affine Configurational Theorems
% 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 : pardepap [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 6 ( 1 unt; 0 def)
% Number of atoms : 125 ( 0 equ)
% Maximal formula atoms : 52 ( 20 avg)
% Number of connectives : 133 ( 14 ~; 8 |; 51 &)
% ( 0 <=>; 60 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 19 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 11 ( 10 usr; 1 prp; 0-5 aty)
% Number of functors : 1 ( 1 usr; 0 con; 1-1 aty)
% Number of variables : 53 ( 51 !; 2 ?)
% 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(t1_pardepap,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v2_aff_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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( r2_analoaf(A,B,C,B,D)
& r2_analoaf(A,E,F,E,G)
& r2_analoaf(A,B,F,C,E)
& r2_analoaf(A,C,G,D,F) )
=> r2_analoaf(A,D,E,B,G) ) ) ) ) ) ) ) ) ) ).
fof(t2_pardepap,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r2_analoaf(A,B,C,B,D)
& r2_analoaf(A,B,E,B,F)
& r2_analoaf(A,B,G,B,H)
& r2_analoaf(A,C,E,D,F)
& r2_analoaf(A,C,G,D,H) )
=> ( r2_analoaf(A,B,C,B,E)
| r2_analoaf(A,B,C,B,G)
| r2_analoaf(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t3_pardepap,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v11_aff_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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( r2_analoaf(A,B,C,D,E)
& r2_analoaf(A,B,C,F,G)
& r2_analoaf(A,B,D,C,E)
& r2_analoaf(A,B,F,C,G) )
=> ( r2_analoaf(A,B,C,B,D)
| r2_analoaf(A,B,C,B,F)
| r2_analoaf(A,D,F,E,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_pardepap,axiom,
$true ).
fof(t5_pardepap,axiom,
? [A] :
( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r2_analoaf(A,B,C,B,D)
& r2_analoaf(A,B,E,B,F)
& r2_analoaf(A,B,G,B,H)
& r2_analoaf(A,C,E,D,F)
& r2_analoaf(A,C,G,D,H) )
=> ( r2_analoaf(A,B,C,B,E)
| r2_analoaf(A,B,C,B,G)
| r2_analoaf(A,E,G,F,H) ) ) ) ) ) ) ) ) )
& ! [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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( r2_analoaf(A,B,C,D,E)
& r2_analoaf(A,B,C,F,G)
& r2_analoaf(A,B,D,C,E)
& r2_analoaf(A,B,F,C,G) )
=> ( r2_analoaf(A,B,C,B,D)
| r2_analoaf(A,B,C,B,F)
| r2_analoaf(A,D,F,E,G) ) ) ) ) ) ) ) )
& ! [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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( r2_analoaf(A,B,C,B,D)
& r2_analoaf(A,E,F,E,G)
& r2_analoaf(A,B,F,C,E)
& r2_analoaf(A,C,G,D,F) )
=> r2_analoaf(A,D,E,B,G) ) ) ) ) ) ) )
& ! [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))
=> ~ ( ~ r2_analoaf(A,B,C,B,D)
& r2_analoaf(A,B,C,D,E)
& r2_analoaf(A,B,D,C,E)
& r2_analoaf(A,B,E,C,D) ) ) ) ) ) ) ).
fof(t6_pardepap,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(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))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r2_analoaf(A,B,C,B,D)
& ~ ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( r2_analoaf(A,B,D,B,E)
& ~ ( r2_analoaf(A,B,F,B,G)
& r2_analoaf(A,D,F,E,G) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------