SET007 Axioms: SET007+13.ax
%------------------------------------------------------------------------------
% File : SET007+13 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Binary Relations
% 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 : relat_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 63 ( 17 unt; 0 def)
% Number of atoms : 200 ( 6 equ)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 146 ( 9 ~; 0 |; 37 &)
% ( 24 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 0 con; 1-2 aty)
% Number of variables : 79 ( 79 !; 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_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_relat_2(A,B)
<=> ! [C] :
( r2_hidden(C,B)
=> r2_hidden(k4_tarski(C,C),A) ) ) ) ).
fof(d2_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_relat_2(A,B)
<=> ! [C] :
~ ( r2_hidden(C,B)
& r2_hidden(k4_tarski(C,C),A) ) ) ) ).
fof(d3_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r3_relat_2(A,B)
<=> ! [C,D] :
( ( r2_hidden(C,B)
& r2_hidden(D,B)
& r2_hidden(k4_tarski(C,D),A) )
=> r2_hidden(k4_tarski(D,C),A) ) ) ) ).
fof(d4_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r4_relat_2(A,B)
<=> ! [C,D] :
( ( r2_hidden(C,B)
& r2_hidden(D,B)
& r2_hidden(k4_tarski(C,D),A)
& r2_hidden(k4_tarski(D,C),A) )
=> C = D ) ) ) ).
fof(d5_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r5_relat_2(A,B)
<=> ! [C,D] :
~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& r2_hidden(k4_tarski(C,D),A)
& r2_hidden(k4_tarski(D,C),A) ) ) ) ).
fof(d6_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r6_relat_2(A,B)
<=> ! [C,D] :
~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& ~ r2_hidden(k4_tarski(C,D),A)
& ~ r2_hidden(k4_tarski(D,C),A) ) ) ) ).
fof(d7_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r7_relat_2(A,B)
<=> ! [C,D] :
~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& ~ r2_hidden(k4_tarski(C,D),A)
& ~ r2_hidden(k4_tarski(D,C),A) ) ) ) ).
fof(d8_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r8_relat_2(A,B)
<=> ! [C,D,E] :
( ( r2_hidden(C,B)
& r2_hidden(D,B)
& r2_hidden(E,B)
& r2_hidden(k4_tarski(C,D),A)
& r2_hidden(k4_tarski(D,E),A) )
=> r2_hidden(k4_tarski(C,E),A) ) ) ) ).
fof(d9_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_relat_2(A)
<=> r1_relat_2(A,k3_relat_1(A)) ) ) ).
fof(d10_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_relat_2(A)
<=> r2_relat_2(A,k3_relat_1(A)) ) ) ).
fof(d11_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_relat_2(A)
<=> r3_relat_2(A,k3_relat_1(A)) ) ) ).
fof(d12_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v4_relat_2(A)
<=> r4_relat_2(A,k3_relat_1(A)) ) ) ).
fof(d13_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v5_relat_2(A)
<=> r5_relat_2(A,k3_relat_1(A)) ) ) ).
fof(d14_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v6_relat_2(A)
<=> r6_relat_2(A,k3_relat_1(A)) ) ) ).
fof(d15_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v7_relat_2(A)
<=> r7_relat_2(A,k3_relat_1(A)) ) ) ).
fof(d16_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v8_relat_2(A)
<=> r8_relat_2(A,k3_relat_1(A)) ) ) ).
fof(t1_relat_2,axiom,
$true ).
fof(t2_relat_2,axiom,
$true ).
fof(t3_relat_2,axiom,
$true ).
fof(t4_relat_2,axiom,
$true ).
fof(t5_relat_2,axiom,
$true ).
fof(t6_relat_2,axiom,
$true ).
fof(t7_relat_2,axiom,
$true ).
fof(t8_relat_2,axiom,
$true ).
fof(t9_relat_2,axiom,
$true ).
fof(t10_relat_2,axiom,
$true ).
fof(t11_relat_2,axiom,
$true ).
fof(t12_relat_2,axiom,
$true ).
fof(t13_relat_2,axiom,
$true ).
fof(t14_relat_2,axiom,
$true ).
fof(t15_relat_2,axiom,
$true ).
fof(t16_relat_2,axiom,
$true ).
fof(t17_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_relat_2(A)
<=> r1_tarski(k6_relat_1(k3_relat_1(A)),A) ) ) ).
fof(t18_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_relat_2(A)
<=> r1_xboole_0(k6_relat_1(k3_relat_1(A)),A) ) ) ).
fof(t19_relat_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r4_relat_2(B,A)
<=> r5_relat_2(k4_xboole_0(B,k6_relat_1(A)),A) ) ) ).
fof(t20_relat_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r5_relat_2(B,A)
=> r4_relat_2(k2_xboole_0(B,k6_relat_1(A)),A) ) ) ).
fof(t21_relat_2,axiom,
$true ).
fof(t22_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( ( v3_relat_2(A)
& v8_relat_2(A) )
=> v1_relat_2(A) ) ) ).
fof(t23_relat_2,axiom,
! [A] :
( v3_relat_2(k6_relat_1(A))
& v8_relat_2(k6_relat_1(A)) ) ).
fof(t24_relat_2,axiom,
! [A] :
( v4_relat_2(k6_relat_1(A))
& v1_relat_2(k6_relat_1(A)) ) ).
fof(t25_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( ( v2_relat_2(A)
& v8_relat_2(A) )
=> v5_relat_2(A) ) ) ).
fof(t26_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v5_relat_2(A)
=> ( v2_relat_2(A)
& v4_relat_2(A) ) ) ) ).
fof(t27_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_relat_2(A)
=> v1_relat_2(k4_relat_1(A)) ) ) ).
fof(t28_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_relat_2(A)
=> v2_relat_2(k4_relat_1(A)) ) ) ).
fof(t29_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_relat_2(A)
=> ( k1_relat_1(A) = k1_relat_1(k4_relat_1(A))
& k2_relat_1(A) = k2_relat_1(k4_relat_1(A)) ) ) ) ).
fof(t30_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_relat_2(A)
<=> A = k4_relat_1(A) ) ) ).
fof(t31_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( ( v1_relat_2(A)
& v1_relat_2(B) )
=> ( v1_relat_2(k2_xboole_0(A,B))
& v1_relat_2(k3_xboole_0(A,B)) ) ) ) ) ).
fof(t32_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( ( v2_relat_2(A)
& v2_relat_2(B) )
=> ( v2_relat_2(k2_xboole_0(A,B))
& v2_relat_2(k3_xboole_0(A,B)) ) ) ) ) ).
fof(t33_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( v2_relat_2(A)
=> v2_relat_2(k4_xboole_0(A,B)) ) ) ) ).
fof(t34_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_relat_2(A)
=> v3_relat_2(k4_relat_1(A)) ) ) ).
fof(t35_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( ( v3_relat_2(A)
& v3_relat_2(B) )
=> ( v3_relat_2(k2_xboole_0(A,B))
& v3_relat_2(k3_xboole_0(A,B))
& v3_relat_2(k4_xboole_0(A,B)) ) ) ) ) ).
fof(t36_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v5_relat_2(A)
=> v5_relat_2(k4_relat_1(A)) ) ) ).
fof(t37_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( ( v5_relat_2(A)
& v5_relat_2(B) )
=> v5_relat_2(k3_xboole_0(A,B)) ) ) ) ).
fof(t38_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( v5_relat_2(A)
=> v5_relat_2(k4_xboole_0(A,B)) ) ) ) ).
fof(t39_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v4_relat_2(A)
<=> r1_tarski(k3_xboole_0(A,k4_relat_1(A)),k6_relat_1(k1_relat_1(A))) ) ) ).
fof(t40_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v4_relat_2(A)
=> v4_relat_2(k4_relat_1(A)) ) ) ).
fof(t41_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( v4_relat_2(A)
=> ( v4_relat_2(k3_xboole_0(A,B))
& v4_relat_2(k4_xboole_0(A,B)) ) ) ) ) ).
fof(t42_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v8_relat_2(A)
=> v8_relat_2(k4_relat_1(A)) ) ) ).
fof(t43_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( ( v8_relat_2(A)
& v8_relat_2(B) )
=> v8_relat_2(k3_xboole_0(A,B)) ) ) ) ).
fof(t44_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v8_relat_2(A)
<=> r1_tarski(k5_relat_1(A,A),A) ) ) ).
fof(t45_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v6_relat_2(A)
<=> r1_tarski(k4_xboole_0(k2_zfmisc_1(k3_relat_1(A),k3_relat_1(A)),k6_relat_1(k3_relat_1(A))),k2_xboole_0(A,k4_relat_1(A))) ) ) ).
fof(t46_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v7_relat_2(A)
=> ( v6_relat_2(A)
& v1_relat_2(A) ) ) ) ).
fof(t47_relat_2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v7_relat_2(A)
<=> k2_zfmisc_1(k3_relat_1(A),k3_relat_1(A)) = k2_xboole_0(A,k4_relat_1(A)) ) ) ).
%------------------------------------------------------------------------------