%------------------------------------------------------------------------------
% File : SET007+285 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Metric-Affine Configurations in Metric Affine Planes - Part II
% 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 : conmetr [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 23 ( 0 unt; 0 def)
% Number of atoms : 361 ( 29 equ)
% Maximal formula atoms : 36 ( 15 avg)
% Number of connectives : 374 ( 36 ~; 46 |; 149 &)
% ( 8 <=>; 135 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 18 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 25 ( 24 usr; 0 prp; 1-5 aty)
% Number of functors : 3 ( 3 usr; 0 con; 1-1 aty)
% Number of variables : 119 ( 116 !; 3 ?)
% 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(d1_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v1_conmetr(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,k1_zfmisc_1(u1_struct_0(A)))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,I)
& r2_hidden(C,I)
& r2_hidden(D,I)
& r2_hidden(E,I)
& r2_hidden(B,J)
& r2_hidden(F,J)
& r2_hidden(G,J)
& r2_hidden(H,J)
& r11_analmetr(A,I,J)
& r4_analmetr(A,E,G,D,F)
& r4_analmetr(A,E,H,C,F) )
=> ( r2_hidden(G,I)
| r2_hidden(E,J)
| B = C
| B = D
| B = E
| B = F
| B = G
| B = H
| r4_analmetr(A,C,G,D,H) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v2_conmetr(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,k1_zfmisc_1(u1_struct_0(A)))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_analmetr(I,A)
& v4_analmetr(J,A)
& r2_hidden(B,I)
& r2_hidden(C,I)
& r2_hidden(D,I)
& r2_hidden(E,I)
& r2_hidden(B,J)
& r2_hidden(F,J)
& r2_hidden(G,J)
& r2_hidden(H,J)
& r4_analmetr(A,E,G,D,F)
& r4_analmetr(A,E,H,C,F) )
=> ( r2_hidden(G,I)
| r2_hidden(E,J)
| B = C
| B = D
| B = E
| B = F
| B = G
| B = H
| r4_analmetr(A,C,G,D,H) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v3_conmetr(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,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r11_analmetr(A,J,K)
& r2_hidden(B,J)
& r2_hidden(D,J)
& r2_hidden(F,J)
& r2_hidden(H,J)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(G,K)
& r2_hidden(I,K)
& r5_analmetr(A,B,C,F,G)
& r5_analmetr(A,C,D,G,H)
& r5_analmetr(A,D,E,H,I) )
=> ( r2_hidden(C,J)
| r2_hidden(E,J)
| r5_analmetr(A,B,E,F,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v4_conmetr(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,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r11_analmetr(A,J,K)
& r2_hidden(B,J)
& r2_hidden(D,J)
& r2_hidden(G,J)
& r2_hidden(I,J)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(F,K)
& r2_hidden(H,K)
& r5_analmetr(A,B,C,F,G)
& r5_analmetr(A,C,D,G,H)
& r5_analmetr(A,D,E,H,I) )
=> ( r2_hidden(C,J)
| r2_hidden(E,J)
| r5_analmetr(A,B,E,F,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v5_conmetr(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))
=> ( ( r6_analmetr(A,B,C,D)
& r6_analmetr(A,B,E,F)
& r6_analmetr(A,B,G,H)
& r4_analmetr(A,C,E,D,F)
& r4_analmetr(A,C,E,B,G)
& r4_analmetr(A,E,G,F,H) )
=> ( B = C
| B = D
| B = E
| B = F
| B = G
| B = H
| r6_analmetr(A,E,F,C)
| r6_analmetr(A,E,F,G)
| r4_analmetr(A,C,G,D,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v6_conmetr(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,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_analmetr(J,A)
& v4_analmetr(K,A)
& r2_hidden(B,J)
& r2_hidden(D,J)
& r2_hidden(F,J)
& r2_hidden(H,J)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(G,K)
& r2_hidden(I,K)
& r4_analmetr(A,D,C,H,G)
& r4_analmetr(A,C,B,G,F)
& r4_analmetr(A,B,E,F,I) )
=> ( r2_hidden(E,J)
| r2_hidden(C,J)
| r2_hidden(G,J)
| r2_hidden(I,J)
| r2_hidden(B,K)
| r2_hidden(D,K)
| r2_hidden(F,K)
| r2_hidden(H,K)
| r4_analmetr(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d7_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v7_conmetr(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,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r11_analmetr(A,J,K)
& r2_hidden(B,J)
& r2_hidden(D,J)
& r2_hidden(F,J)
& r2_hidden(H,J)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(G,K)
& r2_hidden(I,K)
& r4_analmetr(A,D,C,H,G)
& r4_analmetr(A,C,B,G,F)
& r4_analmetr(A,B,E,F,I) )
=> ( r2_hidden(E,J)
| r2_hidden(C,J)
| r2_hidden(G,J)
| r2_hidden(I,J)
| r2_hidden(B,K)
| r2_hidden(D,K)
| r2_hidden(F,K)
| r2_hidden(H,K)
| r4_analmetr(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d8_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v8_conmetr(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))
=> ( ( r4_analmetr(A,B,C,D,E)
& r4_analmetr(A,B,C,F,G)
& r4_analmetr(A,B,D,C,E)
& r4_analmetr(A,B,F,C,G) )
=> ( r6_analmetr(A,B,C,D)
| r6_analmetr(A,B,C,F)
| r4_analmetr(A,D,F,E,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(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))
& r6_analmetr(A,B,C,D)
& B != C
& C != D
& D != B ) ) ) ) ).
fof(t2_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(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))
=> ~ ( r6_analmetr(A,B,C,D)
& B != D
& C != D ) ) ) ) ) ) ).
fof(t3_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( v4_analmetr(B,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(C,D)
& r11_analmetr(A,B,D) ) ) ) ) ) ) ).
fof(t4_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_analmetr(E,A)
& r2_hidden(B,E)
& r2_hidden(C,E)
& r2_hidden(D,E) )
=> r6_analmetr(A,B,C,D) ) ) ) ) ) ) ).
fof(t5_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(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)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v4_analmetr(D,A)
& v4_analmetr(E,A)
& r2_hidden(B,D)
& r2_hidden(C,D)
& r2_hidden(B,E)
& r2_hidden(C,E)
& B != C
& D != E ) ) ) ) ) ) ).
fof(t6_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(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,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k3_analmetr(A))))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k3_analmetr(A)))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(k3_analmetr(A)))
=> ( ( D = H
& E = I
& F = G
& r2_hidden(B,F)
& r2_hidden(C,F)
& r2_aff_1(k3_analmetr(A),H,I,G) )
=> r4_analmetr(A,D,E,B,C) ) ) ) ) ) ) ) ) ) ) ).
fof(t7_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v5_conmetr(A)
=> v7_aff_2(k3_analmetr(A)) ) ) ).
fof(t8_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v11_aff_2(k3_analmetr(A))
=> v8_conmetr(A) ) ) ).
fof(t9_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v3_conmetr(A)
=> v7_conmetr(A) ) ) ).
fof(t10_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v4_conmetr(A)
=> v7_conmetr(A) ) ) ).
fof(t11_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v2_conaffm(A)
=> v5_conmetr(A) ) ) ).
fof(t12_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( ( v7_conmetr(A)
& v5_conmetr(A) )
=> v6_conmetr(A) ) ) ).
fof(t13_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( ( v1_conmetr(A)
& v1_conaffm(A) )
=> v2_conmetr(A) ) ) ).
fof(t14_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( ( v3_conmetr(A)
& v4_conmetr(A) )
=> v1_conmetr(A) ) ) ).
fof(t15_conmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v3_conaffm(A)
=> v1_conmetr(A) ) ) ).
%------------------------------------------------------------------------------