%------------------------------------------------------------------------------
% File : SET007+231 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Affine Localizations of Desargues Axiom
% 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 : aff_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 26 ( 8 unt; 0 def)
% Number of atoms : 426 ( 52 equ)
% Maximal formula atoms : 47 ( 16 avg)
% Number of connectives : 436 ( 36 ~; 68 |; 196 &)
% ( 10 <=>; 126 =>; 0 <=; 0 <~>)
% Maximal formula depth : 45 ( 16 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-5 aty)
% Number of functors : 2 ( 2 usr; 0 con; 1-1 aty)
% Number of variables : 110 ( 110 !; 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(d1_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v1_aff_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(A))
=> ! [M] :
( m1_subset_1(M,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,B)
& r2_hidden(E,C)
& r2_hidden(H,C)
& r2_hidden(I,C)
& r2_hidden(E,D)
& r2_hidden(J,D)
& r2_hidden(K,D)
& r1_aff_1(A,H,F,L)
& r1_aff_1(A,I,G,L)
& r1_aff_1(A,H,J,M)
& r1_aff_1(A,I,K,M)
& r2_analoaf(A,F,J,G,K) )
=> ( C = B
| C = D
| B = D
| E = F
| E = H
| E = J
| L = M
| r1_aff_1(A,H,F,J)
| r1_aff_1(A,I,G,K)
| F = G
| r2_analoaf(A,F,J,L,M) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v2_aff_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(A))
=> ! [M] :
( m1_subset_1(M,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,B)
& r2_hidden(E,C)
& r2_hidden(H,C)
& r2_hidden(I,C)
& r2_hidden(E,D)
& r2_hidden(J,D)
& r2_hidden(K,D)
& r1_aff_1(A,H,F,L)
& r1_aff_1(A,I,G,L)
& r1_aff_1(A,H,J,M)
& r1_aff_1(A,I,K,M)
& r2_analoaf(A,F,J,L,M) )
=> ( C = B
| C = D
| B = D
| E = F
| E = H
| E = J
| L = M
| J = M
| r1_aff_1(A,H,F,J)
| r1_aff_1(A,I,G,K)
| r2_analoaf(A,F,J,G,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v3_aff_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(A))
=> ! [M] :
( m1_subset_1(M,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,B)
& r2_hidden(E,C)
& r2_hidden(H,C)
& r2_hidden(I,C)
& r2_hidden(J,D)
& r2_hidden(K,D)
& r1_aff_1(A,H,F,L)
& r1_aff_1(A,I,G,L)
& r1_aff_1(A,H,J,M)
& r1_aff_1(A,I,K,M)
& r2_analoaf(A,F,J,G,K)
& r2_analoaf(A,F,J,L,M) )
=> ( C = B
| C = D
| B = D
| E = F
| E = H
| E = J
| L = M
| r1_aff_1(A,H,F,J)
| r1_aff_1(A,I,G,K)
| J = K
| r2_hidden(E,D) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(A))
=> ! [M] :
( m1_subset_1(M,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,B)
& r2_hidden(H,C)
& r2_hidden(I,C)
& r2_hidden(E,D)
& r2_hidden(J,D)
& r2_hidden(K,D)
& r1_aff_1(A,H,F,L)
& r1_aff_1(A,I,G,L)
& r1_aff_1(A,H,J,M)
& r1_aff_1(A,I,K,M)
& r2_analoaf(A,F,J,G,K)
& r2_analoaf(A,F,J,L,M) )
=> ( C = B
| C = D
| B = D
| E = F
| E = H
| E = J
| L = M
| r1_aff_1(A,H,F,J)
| r1_aff_1(A,I,G,K)
| H = I
| F = G
| r2_hidden(E,C) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v5_aff_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,C)
& r2_hidden(H,C)
& r2_hidden(I,D)
& r2_hidden(J,D)
& r4_aff_1(A,B,C)
& r4_aff_1(A,B,D)
& r1_aff_1(A,G,E,K)
& r1_aff_1(A,H,F,K)
& r1_aff_1(A,G,I,L)
& r1_aff_1(A,H,J,L)
& r2_analoaf(A,E,I,F,J) )
=> ( B = C
| B = D
| C = D
| r1_aff_1(A,G,E,I)
| r1_aff_1(A,H,F,J)
| K = L
| E = F
| r2_analoaf(A,E,I,K,L) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v6_aff_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,C)
& r2_hidden(H,C)
& r2_hidden(I,D)
& r2_hidden(J,D)
& r4_aff_1(A,B,C)
& r4_aff_1(A,B,D)
& r1_aff_1(A,G,E,K)
& r1_aff_1(A,H,F,K)
& r1_aff_1(A,G,I,L)
& r1_aff_1(A,H,J,L)
& r2_analoaf(A,E,I,K,L) )
=> ( B = C
| B = D
| C = D
| r1_aff_1(A,G,E,I)
| r1_aff_1(A,H,F,J)
| K = L
| r2_analoaf(A,E,I,F,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d7_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v7_aff_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,C)
& r2_hidden(H,C)
& r2_hidden(I,D)
& r2_hidden(J,D)
& r4_aff_1(A,B,D)
& r1_aff_1(A,G,E,K)
& r1_aff_1(A,H,F,K)
& r1_aff_1(A,G,I,L)
& r1_aff_1(A,H,J,L)
& r2_analoaf(A,E,I,F,J)
& r2_analoaf(A,E,I,K,L) )
=> ( B = C
| B = D
| C = D
| r1_aff_1(A,G,E,I)
| r1_aff_1(A,H,F,J)
| K = L
| E = F
| r4_aff_1(A,B,C) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d8_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v8_aff_3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,C)
& r2_hidden(H,C)
& r2_hidden(I,D)
& r2_hidden(J,D)
& r4_aff_1(A,B,C)
& r1_aff_1(A,G,E,K)
& r1_aff_1(A,H,F,K)
& r1_aff_1(A,G,I,L)
& r1_aff_1(A,H,J,L)
& r2_analoaf(A,E,I,F,J)
& r2_analoaf(A,E,I,K,L) )
=> ( B = C
| B = D
| C = D
| r1_aff_1(A,G,E,I)
| r1_aff_1(A,H,F,J)
| K = L
| I = J
| r4_aff_1(A,B,D) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_aff_3,axiom,
$true ).
fof(t2_aff_3,axiom,
$true ).
fof(t3_aff_3,axiom,
$true ).
fof(t4_aff_3,axiom,
$true ).
fof(t5_aff_3,axiom,
$true ).
fof(t6_aff_3,axiom,
$true ).
fof(t7_aff_3,axiom,
$true ).
fof(t8_aff_3,axiom,
$true ).
fof(t9_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v1_aff_3(A)
=> v2_aff_3(A) ) ) ).
fof(t10_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v2_aff_3(A)
=> v1_aff_3(A) ) ) ).
fof(t11_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_2(A)
=> v1_aff_3(A) ) ) ).
fof(t12_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_2(A)
=> v3_aff_3(A) ) ) ).
fof(t13_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v3_aff_3(A)
=> v4_aff_3(A) ) ) ).
fof(t14_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v3_aff_3(A)
=> v4_aff_2(A) ) ) ).
fof(t15_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v6_aff_3(A)
=> v5_aff_3(A) ) ) ).
fof(t16_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v6_aff_3(A)
<=> v8_aff_3(A) ) ) ).
fof(t17_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v5_aff_3(A)
<=> v7_aff_3(A) ) ) ).
fof(t18_aff_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_3(A)
=> v6_aff_3(A) ) ) ).
%------------------------------------------------------------------------------