SET007 Axioms: SET007+232.ax
%------------------------------------------------------------------------------
% File : SET007+232 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Collinearity Structure
% 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 : collsp [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 54 ( 14 unt; 0 def)
% Number of atoms : 346 ( 36 equ)
% Maximal formula atoms : 15 ( 6 avg)
% Number of connectives : 351 ( 59 ~; 7 |; 147 &)
% ( 8 <=>; 130 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-4 aty)
% Number of functors : 11 ( 11 usr; 1 con; 0-6 aty)
% Number of variables : 140 ( 128 !; 12 ?)
% 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(rc1_collsp,axiom,
? [A] :
( l1_collsp(A)
& v1_collsp(A) ) ).
fof(rc2_collsp,axiom,
? [A] :
( l1_collsp(A)
& ~ v3_struct_0(A)
& v1_collsp(A) ) ).
fof(rc3_collsp,axiom,
? [A] :
( l1_collsp(A)
& ~ v3_struct_0(A)
& v1_collsp(A)
& v2_collsp(A)
& v3_collsp(A) ) ).
fof(rc4_collsp,axiom,
? [A] :
( l1_collsp(A)
& ~ v3_struct_0(A)
& v1_collsp(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A) ) ).
fof(d1_collsp,axiom,
! [A,B] :
( m1_collsp(B,A)
<=> r1_tarski(B,k3_zfmisc_1(A,A,A)) ) ).
fof(t1_collsp,axiom,
$true ).
fof(t2_collsp,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
( k1_tarski(B) != A
& ! [C,D] :
~ ( C != D
& r2_hidden(C,A)
& r2_hidden(D,A) ) ) ) ).
fof(d2_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(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)
<=> r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,D),u1_collsp(A)) ) ) ) ) ) ).
fof(d3_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_collsp(A) )
=> ( v2_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))
=> ( ~ ( B != C
& B != D
& C != D )
=> r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,D),u1_collsp(A)) ) ) ) ) ) ) ).
fof(d4_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_collsp(A) )
=> ( v3_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))
=> ( ( r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,D),u1_collsp(A))
& r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,E),u1_collsp(A))
& r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,F),u1_collsp(A)) )
=> ( B = C
| r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),D,E,F),u1_collsp(A)) ) ) ) ) ) ) ) ) ) ).
fof(t3_collsp,axiom,
$true ).
fof(t4_collsp,axiom,
$true ).
fof(t5_collsp,axiom,
$true ).
fof(t6_collsp,axiom,
$true ).
fof(t7_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(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))
=> ( ~ ( B != C
& B != D
& C != D )
=> r1_collsp(A,B,C,D) ) ) ) ) ) ).
fof(t8_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(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,C,F) )
=> ( B = C
| r1_collsp(A,D,E,F) ) ) ) ) ) ) ) ) ).
fof(t9_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(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,B,D)
& r1_collsp(A,B,D,C) ) ) ) ) ) ) ).
fof(t10_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_collsp(A,B,C,B) ) ) ) ).
fof(t11_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(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(t12_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(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,B,D) ) ) ) ) ) ).
fof(t13_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(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) ) ) ) ) ) ).
fof(t14_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(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(t15_collsp,axiom,
$true ).
fof(t16_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(B,k1_collsp(A,B,C))
& r2_hidden(C,k1_collsp(A,B,C)) ) ) ) ) ).
fof(t17_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(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)
<=> r2_hidden(D,k1_collsp(A,B,C)) ) ) ) ) ) ).
fof(d6_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_collsp(A) )
=> ( v4_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(t18_collsp,axiom,
$true ).
fof(t19_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(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(d7_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m2_collsp(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& C != D
& B = k1_collsp(A,C,D) ) ) ) ) ).
fof(t20_collsp,axiom,
$true ).
fof(t21_collsp,axiom,
$true ).
fof(t22_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( B = C
=> k1_collsp(A,B,C) = u1_struct_0(A) ) ) ) ) ).
fof(t23_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m2_collsp(B,A)
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& C != D
& r2_hidden(C,B)
& r2_hidden(D,B) ) ) ) ) ).
fof(t24_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(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] :
( m2_collsp(D,A)
=> ~ ( r2_hidden(B,D)
& r2_hidden(C,D) ) ) ) ) ) ) ).
fof(t25_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(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] :
( m2_collsp(E,A)
=> ( ( r2_hidden(B,E)
& r2_hidden(C,E)
& r2_hidden(D,E) )
=> r1_collsp(A,B,C,D) ) ) ) ) ) ) ).
fof(t26_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m2_collsp(B,A)
=> ! [C] :
( m2_collsp(C,A)
=> ( r1_tarski(B,C)
=> B = C ) ) ) ) ).
fof(t27_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_collsp(D,A)
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D) )
=> ( B = C
| r1_tarski(k1_collsp(A,B,C),D) ) ) ) ) ) ) ).
fof(t28_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_collsp(D,A)
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D) )
=> ( B = C
| k1_collsp(A,B,C) = D ) ) ) ) ) ) ).
fof(t29_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_collsp(D,A)
=> ! [E] :
( m2_collsp(E,A)
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D)
& r2_hidden(B,E)
& r2_hidden(C,E) )
=> ( B = C
| D = E ) ) ) ) ) ) ) ).
fof(t30_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m2_collsp(B,A)
=> ! [C] :
( m2_collsp(C,A)
=> ~ ( B != C
& ~ r1_xboole_0(B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k3_xboole_0(B,C) != k1_struct_0(A,D) ) ) ) ) ) ).
fof(t31_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& k1_collsp(A,B,C) = u1_struct_0(A) ) ) ) ) ).
fof(dt_m1_collsp,axiom,
$true ).
fof(existence_m1_collsp,axiom,
! [A] :
? [B] : m1_collsp(B,A) ).
fof(dt_m2_collsp,axiom,
$true ).
fof(existence_m2_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& v4_collsp(A)
& l1_collsp(A) )
=> ? [B] : m2_collsp(B,A) ) ).
fof(dt_l1_collsp,axiom,
! [A] :
( l1_collsp(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_collsp,axiom,
? [A] : l1_collsp(A) ).
fof(abstractness_v1_collsp,axiom,
! [A] :
( l1_collsp(A)
=> ( v1_collsp(A)
=> A = g1_collsp(u1_struct_0(A),u1_collsp(A)) ) ) ).
fof(dt_k1_collsp,axiom,
$true ).
fof(dt_u1_collsp,axiom,
! [A] :
( l1_collsp(A)
=> m1_collsp(u1_collsp(A),u1_struct_0(A)) ) ).
fof(dt_g1_collsp,axiom,
! [A,B] :
( m1_collsp(B,A)
=> ( v1_collsp(g1_collsp(A,B))
& l1_collsp(g1_collsp(A,B)) ) ) ).
fof(free_g1_collsp,axiom,
! [A,B] :
( m1_collsp(B,A)
=> ! [C,D] :
( g1_collsp(A,B) = g1_collsp(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(d5_collsp,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_collsp(A)
& v3_collsp(A)
& l1_collsp(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_collsp(A,B,C) = a_3_0_collsp(A,B,C) ) ) ) ).
fof(fraenkel_a_3_0_collsp,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_collsp(B)
& v3_collsp(B)
& l1_collsp(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_collsp(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_collsp(B,C,D,E) ) ) ) ).
%------------------------------------------------------------------------------