SET007 Axioms: SET007+14.ax
%------------------------------------------------------------------------------
% File : SET007+14 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Ordinal Numbers
% 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 : ordinal1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 84 ( 23 unt; 0 def)
% Number of atoms : 314 ( 29 equ)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 264 ( 34 ~; 3 |; 116 &)
% ( 14 <=>; 97 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-2 aty)
% Number of functors : 19 ( 19 usr; 6 con; 0-2 aty)
% Number of variables : 141 ( 130 !; 11 ?)
% 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_ordinal1,axiom,
! [A] : ~ v1_xboole_0(k1_ordinal1(A)) ).
fof(cc1_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v1_ordinal1(A)
& v2_ordinal1(A) ) ) ).
fof(cc2_ordinal1,axiom,
! [A] :
( ( v1_ordinal1(A)
& v2_ordinal1(A) )
=> v3_ordinal1(A) ) ).
fof(rc1_ordinal1,axiom,
? [A] :
( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A) ) ).
fof(rc2_ordinal1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A) ) ).
fof(cc3_ordinal1,axiom,
! [A] :
( v1_xboole_0(A)
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A) ) ) ).
fof(fc2_ordinal1,axiom,
( v1_relat_1(k1_xboole_0)
& v3_relat_1(k1_xboole_0)
& v1_funct_1(k1_xboole_0)
& v2_funct_1(k1_xboole_0)
& v1_xboole_0(k1_xboole_0)
& v1_ordinal1(k1_xboole_0)
& v2_ordinal1(k1_xboole_0)
& v3_ordinal1(k1_xboole_0) ) ).
fof(rc3_ordinal1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A) ) ).
fof(fc3_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( ~ v1_xboole_0(k1_ordinal1(A))
& v1_ordinal1(k1_ordinal1(A))
& v2_ordinal1(k1_ordinal1(A))
& v3_ordinal1(k1_ordinal1(A)) ) ) ).
fof(fc4_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v1_ordinal1(k3_tarski(A))
& v2_ordinal1(k3_tarski(A))
& v3_ordinal1(k3_tarski(A)) ) ) ).
fof(rc4_ordinal1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) ) ).
fof(fc5_ordinal1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> ( v1_ordinal1(k1_relat_1(A))
& v2_ordinal1(k1_relat_1(A))
& v3_ordinal1(k1_relat_1(A)) ) ) ).
fof(t1_ordinal1,axiom,
$true ).
fof(t2_ordinal1,axiom,
$true ).
fof(t3_ordinal1,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& r2_hidden(B,C)
& r2_hidden(C,A) ) ).
fof(t4_ordinal1,axiom,
! [A,B,C,D] :
~ ( r2_hidden(A,B)
& r2_hidden(B,C)
& r2_hidden(C,D)
& r2_hidden(D,A) ) ).
fof(t5_ordinal1,axiom,
! [A,B,C,D,E] :
~ ( r2_hidden(A,B)
& r2_hidden(B,C)
& r2_hidden(C,D)
& r2_hidden(D,E)
& r2_hidden(E,A) ) ).
fof(t6_ordinal1,axiom,
! [A,B,C,D,E,F] :
~ ( r2_hidden(A,B)
& r2_hidden(B,C)
& r2_hidden(C,D)
& r2_hidden(D,E)
& r2_hidden(E,F)
& r2_hidden(F,A) ) ).
fof(t7_ordinal1,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& r1_tarski(B,A) ) ).
fof(d1_ordinal1,axiom,
! [A] : k1_ordinal1(A) = k2_xboole_0(A,k1_tarski(A)) ).
fof(t8_ordinal1,axiom,
$true ).
fof(t9_ordinal1,axiom,
$true ).
fof(t10_ordinal1,axiom,
! [A] : r2_hidden(A,k1_ordinal1(A)) ).
fof(t11_ordinal1,axiom,
$true ).
fof(t12_ordinal1,axiom,
! [A,B] :
( k1_ordinal1(A) = k1_ordinal1(B)
=> A = B ) ).
fof(t13_ordinal1,axiom,
! [A,B] :
( r2_hidden(A,k1_ordinal1(B))
<=> ( r2_hidden(A,B)
| A = B ) ) ).
fof(t14_ordinal1,axiom,
! [A] : A != k1_ordinal1(A) ).
fof(d2_ordinal1,axiom,
! [A] :
( v1_ordinal1(A)
<=> ! [B] :
( r2_hidden(B,A)
=> r1_tarski(B,A) ) ) ).
fof(d3_ordinal1,axiom,
! [A] :
( v2_ordinal1(A)
<=> ! [B,C] :
~ ( r2_hidden(B,A)
& r2_hidden(C,A)
& ~ r2_hidden(B,C)
& B != C
& ~ r2_hidden(C,B) ) ) ).
fof(d4_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
<=> ( v1_ordinal1(A)
& v2_ordinal1(A) ) ) ).
fof(t15_ordinal1,axiom,
$true ).
fof(t16_ordinal1,axiom,
$true ).
fof(t17_ordinal1,axiom,
$true ).
fof(t18_ordinal1,axiom,
$true ).
fof(t19_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v1_ordinal1(C)
=> ( ( r2_hidden(C,A)
& r2_hidden(A,B) )
=> r2_hidden(C,B) ) ) ) ) ).
fof(t20_ordinal1,axiom,
$true ).
fof(t21_ordinal1,axiom,
! [A] :
( v1_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_xboole_0(A,B)
=> r2_hidden(A,B) ) ) ) ).
fof(t22_ordinal1,axiom,
! [A] :
( v1_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( r1_tarski(A,B)
& r2_hidden(B,C) )
=> r2_hidden(A,C) ) ) ) ) ).
fof(t23_ordinal1,axiom,
! [A,B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
=> v3_ordinal1(A) ) ) ).
fof(t24_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( ~ r2_hidden(A,B)
& A != B
& ~ r2_hidden(B,A) ) ) ) ).
fof(t25_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> r3_xboole_0(A,B) ) ) ).
fof(t26_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r1_ordinal1(A,B)
| r2_hidden(B,A) ) ) ) ).
fof(t27_ordinal1,axiom,
v3_ordinal1(k1_xboole_0) ).
fof(t28_ordinal1,axiom,
$true ).
fof(t29_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> v3_ordinal1(k1_ordinal1(A)) ) ).
fof(t30_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> v3_ordinal1(k3_tarski(A)) ) ).
fof(t31_ordinal1,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> ( v3_ordinal1(B)
& r1_tarski(B,A) ) )
=> v3_ordinal1(A) ) ).
fof(t32_ordinal1,axiom,
! [A,B] :
( v3_ordinal1(B)
=> ~ ( r1_tarski(A,B)
& A != k1_xboole_0
& ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(C,A)
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,A)
=> r1_ordinal1(C,D) ) ) ) ) ) ) ).
fof(t33_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
<=> r1_ordinal1(k1_ordinal1(A),B) ) ) ) ).
fof(t34_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,k1_ordinal1(B))
<=> r1_ordinal1(A,B) ) ) ) ).
fof(t35_ordinal1,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> v3_ordinal1(B) )
=> v3_ordinal1(k3_tarski(A)) ) ).
fof(t36_ordinal1,axiom,
! [A] :
~ ( ! [B] :
( r2_hidden(B,A)
=> v3_ordinal1(B) )
& ! [B] :
( v3_ordinal1(B)
=> ~ r1_tarski(A,B) ) ) ).
fof(t37_ordinal1,axiom,
! [A] :
~ ! [B] :
( r2_hidden(B,A)
<=> v3_ordinal1(B) ) ).
fof(t38_ordinal1,axiom,
! [A] :
~ ! [B] :
( v3_ordinal1(B)
=> r2_hidden(B,A) ) ).
fof(t39_ordinal1,axiom,
! [A] :
? [B] :
( v3_ordinal1(B)
& ~ r2_hidden(B,A)
& ! [C] :
( v3_ordinal1(C)
=> ( ~ r2_hidden(C,A)
=> r1_ordinal1(B,C) ) ) ) ).
fof(d5_ordinal1,axiom,
$true ).
fof(d6_ordinal1,axiom,
! [A] :
( v4_ordinal1(A)
<=> A = k3_tarski(A) ) ).
fof(t40_ordinal1,axiom,
$true ).
fof(t41_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v4_ordinal1(A)
<=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(B,A)
=> r2_hidden(k1_ordinal1(B),A) ) ) ) ) ).
fof(t42_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( ~ ( ~ v4_ordinal1(A)
& ! [B] :
( v3_ordinal1(B)
=> A != k1_ordinal1(B) ) )
& ~ ( ? [B] :
( v3_ordinal1(B)
& A = k1_ordinal1(B) )
& v4_ordinal1(A) ) ) ) ).
fof(d7_ordinal1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v5_ordinal1(A)
<=> v3_ordinal1(k1_relat_1(A)) ) ) ).
fof(d8_ordinal1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> ( m1_ordinal1(B,A)
<=> r1_tarski(k2_relat_1(B),A) ) ) ).
fof(t43_ordinal1,axiom,
$true ).
fof(t44_ordinal1,axiom,
$true ).
fof(t45_ordinal1,axiom,
! [A] : m1_ordinal1(k1_xboole_0,A) ).
fof(t46_ordinal1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v3_ordinal1(k1_relat_1(A))
=> m1_ordinal1(A,k2_relat_1(A)) ) ) ).
fof(t47_ordinal1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> ! [C] :
( m1_ordinal1(C,A)
=> m1_ordinal1(C,B) ) ) ).
fof(t48_ordinal1,axiom,
! [A,B] :
( m1_ordinal1(B,A)
=> ! [C] :
( v3_ordinal1(C)
=> m1_ordinal1(k2_ordinal1(B,C),A) ) ) ).
fof(d9_ordinal1,axiom,
! [A] :
( v6_ordinal1(A)
<=> ! [B,C] :
( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r3_xboole_0(B,C) ) ) ).
fof(t49_ordinal1,axiom,
! [A] :
( ( ! [B] :
( r2_hidden(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) ) )
& v6_ordinal1(A) )
=> ( v1_relat_1(k3_tarski(A))
& v1_funct_1(k3_tarski(A))
& v5_ordinal1(k3_tarski(A)) ) ) ).
fof(t50_ordinal1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( ~ r2_xboole_0(A,B)
& A != B
& ~ r2_xboole_0(B,A) ) ) ) ).
fof(s1_ordinal1,axiom,
( ? [A] :
( v3_ordinal1(A)
& p1_s1_ordinal1(A) )
=> ? [A] :
( v3_ordinal1(A)
& p1_s1_ordinal1(A)
& ! [B] :
( v3_ordinal1(B)
=> ( p1_s1_ordinal1(B)
=> r1_ordinal1(A,B) ) ) ) ) ).
fof(s2_ordinal1,axiom,
( ! [A] :
( v3_ordinal1(A)
=> ( ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(B,A)
=> p1_s2_ordinal1(B) ) )
=> p1_s2_ordinal1(A) ) )
=> ! [A] :
( v3_ordinal1(A)
=> p1_s2_ordinal1(A) ) ) ).
fof(s3_ordinal1,axiom,
( ( k1_relat_1(f3_s3_ordinal1) = f1_s3_ordinal1
& ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> ( ( r2_hidden(A,f1_s3_ordinal1)
& B = k2_ordinal1(f3_s3_ordinal1,A) )
=> k1_funct_1(f3_s3_ordinal1,A) = f2_s3_ordinal1(B) ) ) )
& k1_relat_1(f4_s3_ordinal1) = f1_s3_ordinal1
& ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> ( ( r2_hidden(A,f1_s3_ordinal1)
& B = k2_ordinal1(f4_s3_ordinal1,A) )
=> k1_funct_1(f4_s3_ordinal1,A) = f2_s3_ordinal1(B) ) ) ) )
=> f3_s3_ordinal1 = f4_s3_ordinal1 ) ).
fof(s4_ordinal1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& k1_relat_1(A) = f1_s4_ordinal1
& ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C) )
=> ( ( r2_hidden(B,f1_s4_ordinal1)
& C = k2_ordinal1(A,B) )
=> k1_funct_1(A,B) = f2_s4_ordinal1(C) ) ) ) ) ).
fof(s5_ordinal1,axiom,
( ( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( B = f2_s5_ordinal1(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& B = f3_s5_ordinal1(C)
& k1_relat_1(C) = A
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,A)
=> k1_funct_1(C,D) = f3_s5_ordinal1(k2_ordinal1(C,D)) ) ) ) ) )
& ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(A,k1_relat_1(f1_s5_ordinal1))
=> k1_funct_1(f1_s5_ordinal1,A) = f2_s5_ordinal1(A) ) ) )
=> ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(A,k1_relat_1(f1_s5_ordinal1))
=> k1_funct_1(f1_s5_ordinal1,A) = f3_s5_ordinal1(k2_ordinal1(f1_s5_ordinal1,A)) ) ) ) ).
fof(dt_m1_ordinal1,axiom,
! [A,B] :
( m1_ordinal1(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) ) ) ).
fof(existence_m1_ordinal1,axiom,
! [A] :
? [B] : m1_ordinal1(B,A) ).
fof(reflexivity_r1_ordinal1,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> r1_ordinal1(A,A) ) ).
fof(connectedness_r1_ordinal1,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> ( r1_ordinal1(A,B)
| r1_ordinal1(B,A) ) ) ).
fof(redefinition_r1_ordinal1,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> ( r1_ordinal1(A,B)
<=> r1_tarski(A,B) ) ) ).
fof(dt_k1_ordinal1,axiom,
$true ).
fof(dt_k2_ordinal1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v3_ordinal1(B) )
=> m1_ordinal1(k2_ordinal1(A,B),k2_relat_1(A)) ) ).
fof(redefinition_k2_ordinal1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v3_ordinal1(B) )
=> k2_ordinal1(A,B) = k7_relat_1(A,B) ) ).
%------------------------------------------------------------------------------