SET007 Axioms: SET007+69.ax
%------------------------------------------------------------------------------
% File : SET007+69 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Increasing and Continuous Ordinal Sequences
% 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 : ordinal4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 65 ( 3 unt; 0 def)
% Number of atoms : 490 ( 50 equ)
% Maximal formula atoms : 22 ( 7 avg)
% Number of connectives : 462 ( 37 ~; 8 |; 238 &)
% ( 3 <=>; 176 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-2 aty)
% Number of functors : 31 ( 31 usr; 4 con; 0-3 aty)
% Number of variables : 160 ( 153 !; 7 ?)
% 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_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ( v1_ordinal1(k1_ordinal2(A))
& v2_ordinal1(k1_ordinal2(A))
& v3_ordinal1(k1_ordinal2(A)) ) ) ).
fof(fc2_ordinal4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( v1_relat_1(k1_ordinal4(A,B))
& v1_funct_1(k1_ordinal4(A,B))
& v5_ordinal1(k1_ordinal4(A,B))
& v1_ordinal2(k1_ordinal4(A,B)) ) ) ).
fof(rc1_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ? [B] :
( m1_ordinal4(B,A)
& v1_ordinal1(B)
& v2_ordinal1(B)
& v3_ordinal1(B)
& ~ v1_xboole_0(B) ) ) ).
fof(t1_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( k1_relat_1(A) = k1_ordinal1(B)
=> ( r1_ordinal2(k1_ordinal2(A),A)
& k12_ordinal2(A) = k1_ordinal2(A) ) ) ) ) ).
fof(d1_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C) )
=> ( C = k1_ordinal4(A,B)
<=> ( k1_relat_1(C) = k14_ordinal2(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,k1_relat_1(A))
=> k1_funct_1(C,D) = k1_funct_1(A,D) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,k14_ordinal2(k1_relat_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ).
fof(t2_ordinal4,axiom,
$true ).
fof(t3_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal2(C,A)
=> r1_ordinal2(C,k1_ordinal4(B,A)) ) ) ) ) ).
fof(t4_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal2(B,A)
=> r1_ordinal2(k14_ordinal2(C,B),k1_ordinal3(C,A)) ) ) ) ) ).
fof(t5_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal2(B,A)
=> r1_ordinal2(k15_ordinal2(B,C),k4_ordinal3(C,A)) ) ) ) ) ).
fof(t6_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& r1_ordinal2(C,A)
& r1_ordinal2(D,B) )
=> ( ( ? [E] :
( v3_ordinal1(E)
& r2_hidden(E,k1_relat_1(A))
& ~ r1_ordinal1(k1_funct_1(A,E),k1_funct_1(B,E)) )
& ? [E] :
( v3_ordinal1(E)
& r2_hidden(E,k1_relat_1(A))
& ~ r2_hidden(k1_funct_1(A,E),k1_funct_1(B,E)) ) )
| r1_ordinal1(C,D) ) ) ) ) ) ) ).
fof(t7_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& v1_ordinal2(D) )
=> ( ( k1_relat_1(C) = k1_relat_1(A)
& k1_relat_1(A) = k1_relat_1(D)
& r1_ordinal2(B,C)
& r1_ordinal2(B,D)
& ! [E] :
( v3_ordinal1(E)
=> ( r2_hidden(E,k1_relat_1(A))
=> ( r1_ordinal1(k1_funct_1(C,E),k1_funct_1(A,E))
& r1_ordinal1(k1_funct_1(A,E),k1_funct_1(D,E)) ) ) ) )
=> r1_ordinal2(B,A) ) ) ) ) ) ).
fof(t8_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ( ( v4_ordinal1(k1_relat_1(A))
& v2_ordinal2(A) )
=> ( k1_relat_1(A) = k1_xboole_0
| ( r1_ordinal2(k8_ordinal2(A),A)
& k12_ordinal2(A) = k8_ordinal2(A) ) ) ) ) ).
fof(t9_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( v2_ordinal2(A)
& r1_ordinal1(B,C)
& r2_hidden(C,k1_relat_1(A)) )
=> r1_ordinal1(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ) ).
fof(t10_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( ( v2_ordinal2(A)
& r2_hidden(B,k1_relat_1(A)) )
=> r1_ordinal1(B,k1_funct_1(A,B)) ) ) ) ).
fof(t11_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( v2_ordinal2(A)
=> v3_ordinal1(k10_relat_1(A,B)) ) ) ) ).
fof(t12_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( v2_ordinal2(A)
=> ( v1_relat_1(k5_relat_1(A,B))
& v1_funct_1(k5_relat_1(A,B))
& v5_ordinal1(k5_relat_1(A,B))
& v1_ordinal2(k5_relat_1(A,B)) ) ) ) ) ).
fof(t13_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ~ ( v2_ordinal2(A)
& v2_ordinal2(B)
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ~ ( C = k5_relat_1(B,A)
& v2_ordinal2(C) ) ) ) ) ) ).
fof(t14_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& v1_ordinal2(D) )
=> ( ( v2_ordinal2(C)
& r1_ordinal2(B,D)
& k7_ordinal2(k2_relat_1(C)) = k1_relat_1(D)
& A = k5_relat_1(C,D) )
=> r1_ordinal2(B,A) ) ) ) ) ) ).
fof(t15_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( v2_ordinal2(A)
=> v2_ordinal2(k2_ordinal1(A,B)) ) ) ) ).
fof(t16_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ( ( v2_ordinal2(A)
& v4_ordinal1(k1_relat_1(A)) )
=> v4_ordinal1(k8_ordinal2(A)) ) ) ).
fof(t17_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ( ( v2_ordinal2(A)
& v3_ordinal2(A)
& v3_ordinal2(B)
& C = k5_relat_1(A,B) )
=> v3_ordinal2(C) ) ) ) ) ).
fof(t18_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(A,C) = k14_ordinal2(B,C) ) )
=> v2_ordinal2(A) ) ) ) ).
fof(t19_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(A,C) = k15_ordinal2(C,B) ) )
=> ( B = k1_xboole_0
| v2_ordinal2(A) ) ) ) ) ).
fof(t20_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ( A != k1_xboole_0
=> k16_ordinal2(k1_xboole_0,A) = k1_xboole_0 ) ) ).
fof(t21_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( v4_ordinal1(A)
=> ( A = k1_xboole_0
| ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ( ( k1_relat_1(C) = A
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,A)
=> k1_funct_1(C,D) = k16_ordinal2(B,D) ) ) )
=> r1_ordinal2(k16_ordinal2(B,A),C) ) ) ) ) ) ) ).
fof(t22_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( A != k1_xboole_0
& k16_ordinal2(A,B) = k1_xboole_0 ) ) ) ).
fof(t23_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(k4_ordinal2,A)
=> r2_hidden(k16_ordinal2(A,B),k16_ordinal2(A,k1_ordinal1(B))) ) ) ) ).
fof(t24_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( r2_hidden(k4_ordinal2,A)
& r2_hidden(B,C) )
=> r2_hidden(k16_ordinal2(A,B),k16_ordinal2(A,C)) ) ) ) ) ).
fof(t25_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( ( r2_hidden(k4_ordinal2,B)
& ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(A,C) = k16_ordinal2(B,C) ) ) )
=> v2_ordinal2(A) ) ) ) ).
fof(t26_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( ( r2_hidden(k4_ordinal2,A)
& v4_ordinal1(B) )
=> ( B = k1_xboole_0
| ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ( ( k1_relat_1(C) = B
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,B)
=> k1_funct_1(C,D) = k16_ordinal2(A,D) ) ) )
=> k16_ordinal2(A,B) = k8_ordinal2(C) ) ) ) ) ) ) ).
fof(t27_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(B,C)
=> ( A = k1_xboole_0
| r1_ordinal1(k16_ordinal2(A,B),k16_ordinal2(A,C)) ) ) ) ) ) ).
fof(t28_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(A,B)
=> r1_ordinal1(k16_ordinal2(A,C),k16_ordinal2(B,C)) ) ) ) ) ).
fof(t29_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(k4_ordinal2,A)
=> ( B = k1_xboole_0
| r2_hidden(k4_ordinal2,k16_ordinal2(A,B)) ) ) ) ) ).
fof(t30_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> k16_ordinal2(A,k14_ordinal2(B,C)) = k15_ordinal2(k16_ordinal2(A,C),k16_ordinal2(A,B)) ) ) ) ).
fof(t31_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> k16_ordinal2(k16_ordinal2(A,B),C) = k16_ordinal2(A,k15_ordinal2(C,B)) ) ) ) ).
fof(t32_ordinal4,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(k4_ordinal2,A)
=> r1_ordinal1(B,k16_ordinal2(A,B)) ) ) ) ).
fof(d2_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( m1_ordinal4(B,A)
<=> r2_hidden(B,A) ) ) ) ).
fof(d3_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( m2_ordinal4(B,A)
<=> ( k1_relat_1(B) = k2_ordinal2(A)
& r1_tarski(k2_relat_1(B),k2_ordinal2(A)) ) ) ) ) ).
fof(d4_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> k2_ordinal4(A) = k1_xboole_0 ) ).
fof(d5_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> k3_ordinal4(A) = k4_ordinal2 ) ).
fof(t33_ordinal4,axiom,
$true ).
fof(t34_ordinal4,axiom,
$true ).
fof(t35_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( k2_ordinal4(A) = k1_xboole_0
& k3_ordinal4(A) = k4_ordinal2 ) ) ).
fof(t36_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( m2_ordinal4(C,A)
=> r2_hidden(B,k1_relat_1(C)) ) ) ) ).
fof(t37_ordinal4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ( ( r2_hidden(k1_relat_1(A),B)
& r1_tarski(k2_relat_1(A),B) )
=> r2_hidden(k8_ordinal2(A),B) ) ) ) ).
fof(t38_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m2_ordinal4(B,A)
=> ~ ( v2_ordinal2(B)
& v3_ordinal2(B)
& r2_hidden(k5_ordinal2,A)
& ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(C,k1_relat_1(B))
& k1_funct_1(B,C) = C ) ) ) ) ) ).
fof(s1_ordinal4,axiom,
( ( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
=> r2_hidden(f1_s1_ordinal4(A),f1_s1_ordinal4(B)) ) ) )
& ! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( v4_ordinal1(B)
=> ( B = k1_xboole_0
| ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ( ( k1_relat_1(A) = B
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,B)
=> k1_funct_1(C,D) = f1_s1_ordinal4(D) ) ) )
=> r1_ordinal2(f1_s1_ordinal4(B),C) ) ) ) ) ) ) )
=> ? [A] :
( v3_ordinal1(A)
& f1_s1_ordinal4(A) = A ) ) ).
fof(s2_ordinal4,axiom,
? [A] :
( m2_ordinal4(A,f1_s2_ordinal4)
& ! [B] :
( m1_ordinal4(B,f1_s2_ordinal4)
=> k1_funct_1(A,B) = f2_s2_ordinal4(B) ) ) ).
fof(dt_m1_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> v3_ordinal1(B) ) ) ).
fof(existence_m1_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ? [B] : m1_ordinal4(B,A) ) ).
fof(dt_m2_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m2_ordinal4(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) ) ) ) ).
fof(existence_m2_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ? [B] : m2_ordinal4(B,A) ) ).
fof(dt_k1_ordinal4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> ( v1_relat_1(k1_ordinal4(A,B))
& v1_funct_1(k1_ordinal4(A,B))
& v5_ordinal1(k1_ordinal4(A,B)) ) ) ).
fof(dt_k2_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> m1_ordinal4(k2_ordinal4(A),A) ) ).
fof(dt_k3_ordinal4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( ~ v1_xboole_0(k3_ordinal4(A))
& m1_ordinal4(k3_ordinal4(A),A) ) ) ).
fof(dt_k4_ordinal4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m2_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> m1_ordinal4(k4_ordinal4(A,B,C),A) ) ).
fof(redefinition_k4_ordinal4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m2_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> k4_ordinal4(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k5_ordinal4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m2_ordinal4(B,A)
& m2_ordinal4(C,A) )
=> m2_ordinal4(k5_ordinal4(A,B,C),A) ) ).
fof(redefinition_k5_ordinal4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m2_ordinal4(B,A)
& m2_ordinal4(C,A) )
=> k5_ordinal4(A,B,C) = k5_relat_1(B,C) ) ).
fof(dt_k6_ordinal4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A) )
=> ( ~ v1_xboole_0(k6_ordinal4(A,B))
& m1_ordinal4(k6_ordinal4(A,B),A) ) ) ).
fof(redefinition_k6_ordinal4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A) )
=> k6_ordinal4(A,B) = k1_ordinal1(B) ) ).
fof(dt_k7_ordinal4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> m1_ordinal4(k7_ordinal4(A,B,C),A) ) ).
fof(redefinition_k7_ordinal4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> k7_ordinal4(A,B,C) = k14_ordinal2(B,C) ) ).
fof(dt_k8_ordinal4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> m1_ordinal4(k8_ordinal4(A,B,C),A) ) ).
fof(redefinition_k8_ordinal4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> k8_ordinal4(A,B,C) = k15_ordinal2(B,C) ) ).
%------------------------------------------------------------------------------