SET007 Axioms: SET007+267.ax
%------------------------------------------------------------------------------
% File : SET007+267 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Hessenberg 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 : hessenbe [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 22 ( 4 unt; 0 def)
% Number of atoms : 347 ( 38 equ)
% Maximal formula atoms : 38 ( 15 avg)
% Number of connectives : 376 ( 51 ~; 26 |; 184 &)
% ( 0 <=>; 115 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 15 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 14 ( 12 usr; 1 prp; 0-4 aty)
% Number of functors : 1 ( 1 usr; 0 con; 1-1 aty)
% Number of variables : 106 ( 106 !; 0 ?)
% 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_hessenbe,axiom,
! [A] :
( l1_collsp(A)
=> ( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& v6_anproj_2(A) )
=> ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& v5_anproj_2(A) ) ) ) ).
fof(t1_hessenbe,axiom,
$true ).
fof(t2_hessenbe,axiom,
$true ).
fof(t3_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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_collsp(A,B,C,D)
=> ( r1_collsp(A,C,D,B)
& r1_collsp(A,D,B,C)
& r1_collsp(A,C,B,D)
& r1_collsp(A,B,D,C)
& r1_collsp(A,D,C,B) ) ) ) ) ) ) ).
fof(t4_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ( ( r1_collsp(A,B,C,D)
& r1_collsp(A,B,C,E) )
=> ( B = C
| r1_collsp(A,B,D,E) ) ) ) ) ) ) ) ).
fof(t5_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ( ( r1_collsp(A,D,E,B)
& r1_collsp(A,D,E,C)
& r1_collsp(A,B,C,F) )
=> ( B = C
| r1_collsp(A,D,E,F) ) ) ) ) ) ) ) ) ).
fof(t6_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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,u1_struct_0(A))
=> r1_collsp(A,B,C,D) ) ) ) ) ) ).
fof(t7_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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_collsp(A,B,C,D) ) ) ) ) ).
fof(t8_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ~ ( ~ r1_collsp(A,B,C,D)
& r1_collsp(A,B,C,E)
& B != E
& r1_collsp(A,B,E,D) ) ) ) ) ) ) ).
fof(t9_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ( ( r1_collsp(A,B,C,E)
& r1_collsp(A,B,D,E) )
=> ( r1_collsp(A,B,C,D)
| B = E ) ) ) ) ) ) ) ).
fof(t10_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ~ ( ~ r1_collsp(A,B,C,D)
& r1_collsp(A,B,D,E)
& r1_collsp(A,C,D,F)
& D != E
& r1_collsp(A,G,E,F)
& r1_collsp(A,B,C,G)
& B != G
& F = D ) ) ) ) ) ) ) ) ).
fof(t11_hessenbe,axiom,
$true ).
fof(t12_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ( ( r1_collsp(A,B,C,E)
& r1_collsp(A,C,D,F)
& r1_collsp(A,B,D,G)
& r1_collsp(A,F,E,G) )
=> ( r1_collsp(A,B,C,D)
| E = B
| E = C
| F = C
| F = D
| ( B != G
& D != G ) ) ) ) ) ) ) ) ) ) ).
fof(t13_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ~ ( ~ r1_collsp(A,B,C,D)
& r1_collsp(A,B,C,E)
& r1_collsp(A,D,F,E)
& F != D
& E != B
& r1_collsp(A,F,B,D) ) ) ) ) ) ) ) ).
fof(t14_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ~ ( ~ r1_collsp(A,B,C,D)
& r1_collsp(A,B,C,E)
& r1_collsp(A,D,E,F)
& B != E
& E != F
& r1_collsp(A,C,B,F) ) ) ) ) ) ) ) ).
fof(t15_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ~ ( ~ r1_collsp(A,B,C,D)
& r1_collsp(A,B,C,E)
& r1_collsp(A,D,F,E)
& E != F
& C != E
& r1_collsp(A,E,C,F) ) ) ) ) ) ) ) ).
fof(t16_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ~ ( ~ r1_collsp(A,B,C,D)
& r1_collsp(A,B,C,E)
& r1_collsp(A,D,F,B)
& B != E
& B != F
& r1_collsp(A,E,B,F) ) ) ) ) ) ) ) ).
fof(t17_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ( ( r1_collsp(A,D,E,F)
& r1_collsp(A,D,E,G)
& r1_collsp(A,B,C,F)
& r1_collsp(A,B,C,G) )
=> ( B = C
| D = E
| r1_collsp(A,B,C,D)
| F = G ) ) ) ) ) ) ) ) ) ).
fof(t18_hessenbe,axiom,
$true ).
fof(t19_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& l1_collsp(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))
=> ~ ( ~ r1_collsp(A,B,C,D)
& r1_collsp(A,B,C,E)
& r1_collsp(A,B,D,F)
& B != E
& B != F
& r1_collsp(A,B,E,F) ) ) ) ) ) ) ) ).
fof(t20_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& v6_anproj_2(A)
& v7_anproj_2(A)
& l1_collsp(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))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(A))
=> ( ( r1_collsp(A,D,B,C)
& r1_collsp(A,G,E,F)
& r1_collsp(A,D,E,H)
& r1_collsp(A,G,B,H)
& r1_collsp(A,D,F,I)
& r1_collsp(A,C,G,I)
& r1_collsp(A,B,F,J)
& r1_collsp(A,C,E,J) )
=> ( B = C
| D = C
| E = F
| G = E
| G = F
| r1_collsp(A,D,B,G)
| r1_collsp(A,J,I,H) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t21_hessenbe,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& v2_anproj_2(A)
& v3_anproj_2(A)
& v6_anproj_2(A)
& v7_anproj_2(A)
& l1_collsp(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))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(A))
=> ! [K] :
( m1_subset_1(K,u1_struct_0(A))
=> ( ( r1_collsp(A,D,F,I)
& r1_collsp(A,C,E,I)
& r1_collsp(A,F,H,J)
& r1_collsp(A,E,G,J)
& r1_collsp(A,D,H,K)
& r1_collsp(A,C,G,K)
& r1_collsp(A,B,D,C)
& r1_collsp(A,B,F,E)
& r1_collsp(A,B,H,G) )
=> ( B = C
| D = C
| B = E
| F = E
| B = G
| H = G
| r1_collsp(A,B,D,F)
| r1_collsp(A,B,D,H)
| r1_collsp(A,B,F,H)
| r1_collsp(A,J,K,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------