SET007 Axioms: SET007+286.ax
%------------------------------------------------------------------------------
% File : SET007+286 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Fanoian, Pappian and Desarguesian Affine Spaces
% 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 : papdesaf [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 50 ( 15 unt; 0 def)
% Number of atoms : 426 ( 23 equ)
% Maximal formula atoms : 26 ( 8 avg)
% Number of connectives : 468 ( 92 ~; 8 |; 246 &)
% ( 10 <=>; 112 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-5 aty)
% Number of functors : 12 ( 12 usr; 2 con; 0-3 aty)
% Number of variables : 97 ( 92 !; 5 ?)
% 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_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( ~ v3_struct_0(k2_diraf(A))
& ~ v3_realset2(k2_diraf(A))
& v1_analoaf(k2_diraf(A))
& v1_diraf(k2_diraf(A)) ) ) ).
fof(fc2_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& v3_analoaf(A)
& l1_analoaf(A) )
=> ( ~ v3_struct_0(k2_diraf(A))
& ~ v3_realset2(k2_diraf(A))
& v1_analoaf(k2_diraf(A))
& v1_diraf(k2_diraf(A))
& v2_diraf(k2_diraf(A)) ) ) ).
fof(rc1_papdesaf,axiom,
? [A] :
( l1_analoaf(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& v1_papdesaf(A)
& v2_papdesaf(A)
& v3_papdesaf(A)
& v4_papdesaf(A) ) ).
fof(rc2_papdesaf,axiom,
? [A] :
( l1_analoaf(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_analoaf(A)
& v1_diraf(A)
& v2_diraf(A)
& v2_aff_2(A)
& v4_aff_2(A)
& v7_aff_2(A)
& v11_aff_2(A)
& v1_translac(A) ) ).
fof(rc3_papdesaf,axiom,
? [A] :
( l1_analoaf(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_analoaf(A)
& v1_diraf(A)
& v2_aff_2(A)
& v4_aff_2(A)
& v7_aff_2(A)
& v11_aff_2(A)
& v1_translac(A) ) ).
fof(fc3_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( ~ v3_struct_0(k2_diraf(A))
& ~ v3_realset2(k2_diraf(A))
& v1_analoaf(k2_diraf(A))
& v1_diraf(k2_diraf(A))
& v1_translac(k2_diraf(A)) ) ) ).
fof(fc4_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& v1_papdesaf(A)
& l1_analoaf(A) )
=> ( ~ v3_struct_0(k2_diraf(A))
& ~ v3_realset2(k2_diraf(A))
& v1_analoaf(k2_diraf(A))
& v1_diraf(k2_diraf(A))
& v2_aff_2(k2_diraf(A))
& v1_translac(k2_diraf(A)) ) ) ).
fof(fc5_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& v2_papdesaf(A)
& l1_analoaf(A) )
=> ( ~ v3_struct_0(k2_diraf(A))
& ~ v3_realset2(k2_diraf(A))
& v1_analoaf(k2_diraf(A))
& v1_diraf(k2_diraf(A))
& v4_aff_2(k2_diraf(A))
& v1_translac(k2_diraf(A)) ) ) ).
fof(fc6_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& v3_papdesaf(A)
& l1_analoaf(A) )
=> ( ~ v3_struct_0(k2_diraf(A))
& ~ v3_realset2(k2_diraf(A))
& v1_analoaf(k2_diraf(A))
& v1_diraf(k2_diraf(A))
& v7_aff_2(k2_diraf(A))
& v1_translac(k2_diraf(A)) ) ) ).
fof(fc7_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& v4_papdesaf(A)
& l1_analoaf(A) )
=> ( ~ v3_struct_0(k2_diraf(A))
& ~ v3_realset2(k2_diraf(A))
& v1_analoaf(k2_diraf(A))
& v1_diraf(k2_diraf(A))
& v11_aff_2(k2_diraf(A))
& v1_translac(k2_diraf(A)) ) ) ).
fof(t1_papdesaf,axiom,
$true ).
fof(t2_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ! [B] :
( ( m1_subset_1(B,u1_struct_0(A))
=> m1_subset_1(B,u1_struct_0(k2_diraf(A))) )
& ( m1_subset_1(B,u1_struct_0(k2_diraf(A)))
=> m1_subset_1(B,u1_struct_0(A)) )
& ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_diraf(A)))) )
& ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_diraf(A))))
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(t3_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(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(k2_diraf(A)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k2_diraf(A)))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k2_diraf(A)))
=> ( ( B = E
& C = F
& D = G )
=> ( r3_diraf(A,B,C,D)
<=> r1_aff_1(k2_diraf(A),E,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t4_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_analoaf(A)))
<=> m1_subset_1(B,u1_struct_0(A)) ) ) ).
fof(t5_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_analoaf(B)
& l1_analoaf(B) )
=> ( B = k2_analoaf(A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [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))
=> ( ( C = G
& D = H
& E = I
& F = J )
=> ( r2_diraf(B,C,D,E,F)
<=> r1_geomtrap(A,G,H,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t6_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_analoaf(B)
& l1_analoaf(B) )
=> ~ ( B = k2_analoaf(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_numbers)
& ? [F] :
( m1_subset_1(F,k1_numbers)
& k4_rlvect_1(A,k3_rlvect_1(A,C,E),k3_rlvect_1(A,D,F)) = k1_rlvect_1(A)
& ~ ( E = np__0
& F = np__0 ) ) ) ) ) ) ) ) ).
fof(d1_papdesaf,axiom,
$true ).
fof(d2_papdesaf,axiom,
$true ).
fof(d3_papdesaf,axiom,
$true ).
fof(d4_papdesaf,axiom,
$true ).
fof(d5_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v1_translac(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))
=> ( ( r2_analoaf(A,B,C,D,E)
& r2_analoaf(A,B,D,C,E)
& r2_analoaf(A,B,E,C,D) )
=> r2_analoaf(A,B,C,B,D) ) ) ) ) ) ) ) ).
fof(d6_papdesaf,axiom,
$true ).
fof(d7_papdesaf,axiom,
$true ).
fof(d8_papdesaf,axiom,
$true ).
fof(d9_papdesaf,axiom,
$true ).
fof(d10_papdesaf,axiom,
$true ).
fof(d11_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( v1_papdesaf(A)
<=> v2_aff_2(k2_diraf(A)) ) ) ).
fof(d12_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( v2_papdesaf(A)
<=> v4_aff_2(k2_diraf(A)) ) ) ).
fof(d13_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( v3_papdesaf(A)
<=> v7_aff_2(k2_diraf(A)) ) ) ).
fof(d14_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( v4_papdesaf(A)
<=> v11_aff_2(k2_diraf(A)) ) ) ).
fof(d15_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( v5_papdesaf(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,F)
& r2_analoaf(A,B,D,B,G)
& r2_analoaf(A,B,E,B,H)
& r2_analoaf(A,C,D,F,G)
& r2_analoaf(A,C,E,F,H) )
=> ( r3_diraf(A,B,C,D)
| r3_diraf(A,B,C,E)
| r2_analoaf(A,D,E,G,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d16_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( v6_papdesaf(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,C,B,B,F)
& r2_analoaf(A,D,B,B,G)
& r2_analoaf(A,E,B,B,H)
& r2_analoaf(A,C,D,G,F)
& r2_analoaf(A,C,E,H,F) )
=> ( r3_diraf(A,B,C,D)
| r3_diraf(A,B,C,E)
| r2_analoaf(A,D,E,H,G) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t7_papdesaf,axiom,
$true ).
fof(t8_papdesaf,axiom,
$true ).
fof(t9_papdesaf,axiom,
$true ).
fof(t10_papdesaf,axiom,
$true ).
fof(t11_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( v6_papdesaf(A)
=> v5_papdesaf(A) ) ) ).
fof(t12_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(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,C,B,B,E)
& r3_diraf(A,B,D,F)
& r2_diraf(A,C,D,E,F) )
=> ( r3_diraf(A,B,C,D)
| ( r2_analoaf(A,D,B,B,F)
& r2_analoaf(A,C,D,F,E) ) ) ) ) ) ) ) ) ) ).
fof(t13_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(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,E)
& r3_diraf(A,B,D,F)
& r2_diraf(A,C,D,E,F) )
=> ( r3_diraf(A,B,C,D)
| ( r2_analoaf(A,B,D,B,F)
& r2_analoaf(A,C,D,E,F) ) ) ) ) ) ) ) ) ) ).
fof(t14_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> ( v6_papdesaf(A)
=> v4_aff_2(k2_diraf(A)) ) ) ).
fof(t15_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(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,k1_numbers)
=> ( ( k6_rlvect_1(A,B,C) = k3_rlvect_1(A,k6_rlvect_1(A,E,B),G)
& r1_geomtrap(A,B,D,B,F)
& r1_geomtrap(A,C,D,E,F) )
=> ( G = np__0
| r1_geomtrap(A,B,C,B,D)
| ( F = k4_rlvect_1(A,E,k3_rlvect_1(A,k6_rlvect_1(A,D,C),k2_real_1(k1_real_1(G))))
& F = k4_rlvect_1(A,B,k3_rlvect_1(A,k6_rlvect_1(A,D,B),k2_real_1(k1_real_1(G))))
& k6_rlvect_1(A,D,C) = k3_rlvect_1(A,k6_rlvect_1(A,F,E),k1_real_1(G)) ) ) ) ) ) ) ) ) ) ) ).
fof(t16_papdesaf,axiom,
$true ).
fof(t17_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_analoaf(B)
& l1_analoaf(B) )
=> ( B = k2_analoaf(A)
=> v6_papdesaf(B) ) ) ) ).
fof(t18_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_analoaf(B)
& l1_analoaf(B) )
=> ( B = k2_analoaf(A)
=> ( v6_papdesaf(B)
& v5_papdesaf(B) ) ) ) ) ).
fof(t19_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_analoaf(B)
& l1_analoaf(B) )
=> ( B = k2_analoaf(A)
=> v2_aff_2(k2_diraf(B)) ) ) ) ).
fof(t20_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_analoaf(B)
& l1_analoaf(B) )
=> ( B = k2_analoaf(A)
=> v4_aff_2(k2_diraf(B)) ) ) ) ).
fof(t21_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_2(A)
=> v7_aff_2(A) ) ) ).
fof(t22_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_analoaf(B)
& l1_analoaf(B) )
=> ( B = k2_analoaf(A)
=> v7_aff_2(k2_diraf(B)) ) ) ) ).
fof(t23_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v2_analoaf(B)
& l1_analoaf(B) )
=> ( B = k2_analoaf(A)
=> v11_aff_2(k2_diraf(B)) ) ) ) ).
fof(t24_papdesaf,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v2_analoaf(A)
& l1_analoaf(A) )
=> v1_translac(k2_diraf(A)) ) ).
%------------------------------------------------------------------------------