SET007 Axioms: SET007+269.ax
%------------------------------------------------------------------------------
% File : SET007+269 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : One-Dimensional Congruence of Segments, Facts and Midpoint Reln
% 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 : afvect01 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 20 ( 4 unt; 0 def)
% Number of atoms : 238 ( 24 equ)
% Maximal formula atoms : 88 ( 11 avg)
% Number of connectives : 278 ( 60 ~; 2 |; 104 &)
% ( 3 <=>; 109 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 11 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 13 ( 11 usr; 1 prp; 0-5 aty)
% Number of functors : 3 ( 3 usr; 0 con; 1-2 aty)
% Number of variables : 104 ( 100 !; 4 ?)
% 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_afvect01,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_relset_1(B,k2_zfmisc_1(A,A),k2_zfmisc_1(A,A)) )
=> ( ~ v3_struct_0(g1_analoaf(A,B))
& v1_analoaf(g1_analoaf(A,B)) ) ) ).
fof(rc1_afvect01,axiom,
? [A] :
( l1_analoaf(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_analoaf(A)
& v1_afvect01(A) ) ).
fof(d1_afvect01,axiom,
$true ).
fof(d2_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_analoaf(A) )
=> ( v1_afvect01(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_analoaf(A,B,C,C,B) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_analoaf(A,B,C,B,B)
=> B = C ) ) )
& ! [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,F,G)
& r2_analoaf(A,D,E,F,G) )
=> r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) )
& ! [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))
& r2_analoaf(A,B,D,D,C) ) ) )
& ! [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,F,B,F,C)
& r2_analoaf(A,F,D,F,E) )
=> ( B = C
| D = E
| r2_analoaf(A,B,D,C,E) ) ) ) ) ) ) )
& ! [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))
=> ( ~ ( B != D
& r2_analoaf(A,B,C,C,D) )
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( E != F
& r2_analoaf(A,B,C,E,F)
& r2_analoaf(A,B,E,E,C)
& r2_analoaf(A,B,F,F,C) ) ) ) ) ) ) ) )
& ! [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,C,G)
& r2_analoaf(A,C,D,E,F)
& r2_analoaf(A,C,E,E,D)
& r2_analoaf(A,C,F,F,D) )
=> r2_analoaf(A,B,D,D,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))
=> ~ ( B != E
& C != D
& r2_analoaf(A,B,C,C,E)
& r2_analoaf(A,B,D,D,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ~ ( F != G
& r2_analoaf(A,C,D,F,G)
& r2_analoaf(A,C,F,F,D)
& r2_analoaf(A,C,G,G,D) ) ) ) ) ) ) ) )
& ! [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,E,F)
& r2_analoaf(A,B,D,G,H)
& r2_analoaf(A,B,E,E,C)
& r2_analoaf(A,B,G,G,D)
& r2_analoaf(A,B,F,F,C)
& r2_analoaf(A,B,H,H,D)
& ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(A))
=> ~ ( r2_analoaf(A,C,D,I,J)
& r2_analoaf(A,C,I,I,D)
& r2_analoaf(A,C,J,J,D) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_analoaf(A,B,C,B,C) ) ) ) ).
fof(t2_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(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))
=> ( r2_analoaf(A,B,C,D,E)
=> r2_analoaf(A,D,E,B,C) ) ) ) ) ) ) ).
fof(t3_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(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))
=> ( r2_analoaf(A,B,C,D,E)
=> r2_analoaf(A,B,C,E,D) ) ) ) ) ) ) ).
fof(t4_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(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))
=> ( r2_analoaf(A,B,C,D,E)
=> r2_analoaf(A,C,B,D,E) ) ) ) ) ) ) ).
fof(t5_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_analoaf(A,B,B,C,C) ) ) ) ).
fof(t6_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(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))
=> ( r2_analoaf(A,B,C,D,D)
=> B = C ) ) ) ) ) ).
fof(t7_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(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,D,E)
& r2_analoaf(A,B,C,C,F)
& r2_analoaf(A,B,D,D,C)
& r2_analoaf(A,B,E,E,C) )
=> B = F ) ) ) ) ) ) ) ).
fof(t8_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(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))
=> ~ ( r2_analoaf(A,B,C,B,D)
& r2_analoaf(A,B,E,B,D)
& E != C
& E != D
& C != D ) ) ) ) ) ) ).
fof(d3_afvect01,axiom,
$true ).
fof(d4_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_afvect01(A,B,C)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& D != E
& r2_analoaf(A,B,C,D,E)
& r2_analoaf(A,B,D,D,C)
& r2_analoaf(A,B,E,E,C) ) ) ) ) ) ) ).
fof(d5_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(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))
=> ( r2_afvect01(A,B,C,D)
<=> ~ ( ~ ( B = C
& C = D
& B = D )
& ~ ( B = D
& r1_afvect01(A,B,C) )
& ~ ( B != D
& r2_analoaf(A,B,C,C,D) ) ) ) ) ) ) ) ).
fof(t9_afvect01,axiom,
$true ).
fof(t10_afvect01,axiom,
$true ).
fof(t11_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& ~ r1_afvect01(A,B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( B != D
& r2_analoaf(A,B,C,C,D) ) ) ) ) ) ) ).
fof(t12_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(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))
=> ( ( r1_afvect01(A,B,C)
& r2_analoaf(A,B,C,C,D) )
=> B = D ) ) ) ) ) ).
fof(t13_afvect01,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_afvect01(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r1_afvect01(A,B,C)
& B = C ) ) ) ) ).
%------------------------------------------------------------------------------