SET007 Axioms: SET007+27.ax
%------------------------------------------------------------------------------
% File : SET007+27 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Ordinal Arithmetics
% 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 : ordinal3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 101 ( 17 unt; 0 def)
% Number of atoms : 515 ( 110 equ)
% Maximal formula atoms : 19 ( 5 avg)
% Number of connectives : 448 ( 34 ~; 16 |; 135 &)
% ( 9 <=>; 254 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 13 ( 11 usr; 1 prp; 0-2 aty)
% Number of functors : 22 ( 22 usr; 2 con; 0-2 aty)
% Number of variables : 219 ( 218 !; 1 ?)
% 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_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> ( v1_ordinal1(k2_xboole_0(A,B))
& v2_ordinal1(k2_xboole_0(A,B))
& v3_ordinal1(k2_xboole_0(A,B)) ) ) ).
fof(fc2_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> ( v1_ordinal1(k3_xboole_0(A,B))
& v2_ordinal1(k3_xboole_0(A,B))
& v3_ordinal1(k3_xboole_0(A,B)) ) ) ).
fof(t1_ordinal3,axiom,
! [A] : r1_tarski(A,k1_ordinal1(A)) ).
fof(t2_ordinal3,axiom,
! [A,B] :
( r1_tarski(k1_ordinal1(A),B)
=> r1_tarski(A,B) ) ).
fof(t3_ordinal3,axiom,
$true ).
fof(t4_ordinal3,axiom,
$true ).
fof(t5_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
<=> r2_hidden(k1_ordinal1(A),k1_ordinal1(B)) ) ) ) ).
fof(t6_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r1_tarski(B,A)
=> v3_ordinal1(k3_tarski(B)) ) ) ).
fof(t7_ordinal3,axiom,
! [A] : v3_ordinal1(k3_tarski(k2_ordinal2(A))) ).
fof(t8_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r1_tarski(B,A)
=> k2_ordinal2(B) = B ) ) ).
fof(t9_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> k2_ordinal2(k1_tarski(A)) = k1_tarski(A) ) ).
fof(t10_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ( A != k1_xboole_0
=> r2_hidden(k1_xboole_0,A) ) ) ).
fof(t11_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> k6_ordinal2(A) = k1_xboole_0 ) ).
fof(t12_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> k6_ordinal2(k1_tarski(A)) = A ) ).
fof(t13_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r1_tarski(B,A)
=> v3_ordinal1(k1_setfam_1(B)) ) ) ).
fof(t14_ordinal3,axiom,
$true ).
fof(t15_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( k2_xboole_0(A,B) = A
| k2_xboole_0(A,B) = B ) ) ) ).
fof(t16_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( k3_xboole_0(A,B) = A
| k3_xboole_0(A,B) = B ) ) ) ).
fof(t17_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(A,k4_ordinal2)
=> A = k1_xboole_0 ) ) ).
fof(t18_ordinal3,axiom,
k4_ordinal2 = k1_tarski(k1_xboole_0) ).
fof(t19_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ~ ( r1_ordinal1(A,k4_ordinal2)
& A != k1_xboole_0
& A != k4_ordinal2 ) ) ).
fof(t20_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(C,D)
=> ( ( ~ r1_ordinal1(A,B)
& ~ r2_hidden(A,B) )
| r2_hidden(k14_ordinal2(A,C),k14_ordinal2(B,D)) ) ) ) ) ) ) ).
fof(t21_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ( ( r1_ordinal1(A,B)
& r1_ordinal1(C,D) )
=> r1_ordinal1(k14_ordinal2(A,C),k14_ordinal2(B,D)) ) ) ) ) ) ).
fof(t22_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(A,B)
=> ( ( ~ ( r1_ordinal1(C,D)
& D != k1_xboole_0 )
& ~ r2_hidden(C,D) )
| r2_hidden(k15_ordinal2(A,C),k15_ordinal2(B,D)) ) ) ) ) ) ) ).
fof(t23_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ( ( r1_ordinal1(A,B)
& r1_ordinal1(C,D) )
=> r1_ordinal1(k15_ordinal2(A,C),k15_ordinal2(B,D)) ) ) ) ) ) ).
fof(t24_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( k14_ordinal2(A,B) = k14_ordinal2(A,C)
=> B = C ) ) ) ) ).
fof(t25_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(k14_ordinal2(A,B),k14_ordinal2(A,C))
=> r2_hidden(B,C) ) ) ) ) ).
fof(t26_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(k14_ordinal2(A,B),k14_ordinal2(A,C))
=> r1_ordinal1(B,C) ) ) ) ) ).
fof(t27_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r1_ordinal1(A,k14_ordinal2(A,B))
& r1_ordinal1(B,k14_ordinal2(A,B)) ) ) ) ).
fof(t28_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(A,B)
=> ( r2_hidden(A,k14_ordinal2(B,C))
& r2_hidden(A,k14_ordinal2(C,B)) ) ) ) ) ) ).
fof(t29_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( k14_ordinal2(A,B) = k1_xboole_0
=> ( A = k1_xboole_0
& B = k1_xboole_0 ) ) ) ) ).
fof(t30_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( r1_ordinal1(A,B)
& ! [C] :
( v3_ordinal1(C)
=> B != k14_ordinal2(A,C) ) ) ) ) ).
fof(t31_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( r2_hidden(A,B)
& ! [C] :
( v3_ordinal1(C)
=> ~ ( B = k14_ordinal2(A,C)
& C != k1_xboole_0 ) ) ) ) ) ).
fof(t32_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( v4_ordinal1(A)
=> ( A = k1_xboole_0
| v4_ordinal1(k14_ordinal2(B,A)) ) ) ) ) ).
fof(t33_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> k14_ordinal2(k14_ordinal2(A,B),C) = k14_ordinal2(A,k14_ordinal2(B,C)) ) ) ) ).
fof(t34_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( k15_ordinal2(A,B) = k1_xboole_0
& A != k1_xboole_0
& B != k1_xboole_0 ) ) ) ).
fof(t35_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(A,B)
=> ( C = k1_xboole_0
| ( r2_hidden(A,k15_ordinal2(B,C))
& r2_hidden(A,k15_ordinal2(C,B)) ) ) ) ) ) ) ).
fof(t36_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( k15_ordinal2(A,B) = k15_ordinal2(C,B)
=> ( B = k1_xboole_0
| A = C ) ) ) ) ) ).
fof(t37_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(k15_ordinal2(A,B),k15_ordinal2(C,B))
=> r2_hidden(A,C) ) ) ) ) ).
fof(t38_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(k15_ordinal2(A,B),k15_ordinal2(C,B))
=> ( B = k1_xboole_0
| r1_ordinal1(A,C) ) ) ) ) ) ).
fof(t39_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( A != k1_xboole_0
=> ( r1_ordinal1(B,k15_ordinal2(B,A))
& r1_ordinal1(B,k15_ordinal2(A,B)) ) ) ) ) ).
fof(t40_ordinal3,axiom,
$true ).
fof(t41_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( k15_ordinal2(A,B) = k4_ordinal2
=> ( A = k4_ordinal2
& B = k4_ordinal2 ) ) ) ) ).
fof(t42_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(A,k14_ordinal2(B,C))
& ~ r2_hidden(A,B)
& ! [D] :
( v3_ordinal1(D)
=> ~ ( r2_hidden(D,C)
& A = k14_ordinal2(B,D) ) ) ) ) ) ) ).
fof(d1_ordinal3,axiom,
$true ).
fof(d2_ordinal3,axiom,
! [A] :
( v3_ordinal1(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) )
=> ( C = k1_ordinal3(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(B)
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k14_ordinal2(A,k1_funct_1(B,D)) ) ) ) ) ) ) ) ).
fof(d3_ordinal3,axiom,
! [A] :
( v3_ordinal1(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) )
=> ( C = k2_ordinal3(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(B)
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k14_ordinal2(k1_funct_1(B,D),A) ) ) ) ) ) ) ) ).
fof(d4_ordinal3,axiom,
! [A] :
( v3_ordinal1(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) )
=> ( C = k3_ordinal3(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(B)
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k15_ordinal2(A,k1_funct_1(B,D)) ) ) ) ) ) ) ) ).
fof(d5_ordinal3,axiom,
! [A] :
( v3_ordinal1(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) )
=> ( C = k4_ordinal3(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(B)
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k15_ordinal2(k1_funct_1(B,D),A) ) ) ) ) ) ) ) ).
fof(t43_ordinal3,axiom,
$true ).
fof(t44_ordinal3,axiom,
$true ).
fof(t45_ordinal3,axiom,
$true ).
fof(t46_ordinal3,axiom,
$true ).
fof(t47_ordinal3,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)
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( v3_ordinal1(E)
=> ( ( r2_hidden(D,k1_relat_1(A))
& E = k1_funct_1(A,D) )
=> k1_funct_1(B,D) = k14_ordinal2(C,E) ) ) ) )
=> ( k1_xboole_0 = k1_relat_1(A)
| k8_ordinal2(B) = k14_ordinal2(C,k8_ordinal2(A)) ) ) ) ) ) ).
fof(t48_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( v4_ordinal1(A)
=> v4_ordinal1(k15_ordinal2(A,B)) ) ) ) ).
fof(t49_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(A,k15_ordinal2(B,C))
& v4_ordinal1(B)
& ! [D] :
( v3_ordinal1(D)
=> ~ ( r2_hidden(D,B)
& r2_hidden(A,k15_ordinal2(D,C)) ) ) ) ) ) ) ).
fof(t50_ordinal3,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)
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& v4_ordinal1(k8_ordinal2(A))
& ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( v3_ordinal1(E)
=> ( ( r2_hidden(D,k1_relat_1(A))
& E = k1_funct_1(A,D) )
=> k1_funct_1(B,D) = k15_ordinal2(E,C) ) ) ) )
=> ( k1_xboole_0 = k1_relat_1(A)
| C = k1_xboole_0
| k8_ordinal2(B) = k15_ordinal2(k8_ordinal2(A),C) ) ) ) ) ) ).
fof(t51_ordinal3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( k1_xboole_0 != k1_relat_1(A)
=> k8_ordinal2(k1_ordinal3(B,A)) = k14_ordinal2(B,k8_ordinal2(A)) ) ) ) ).
fof(t52_ordinal3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( v4_ordinal1(k8_ordinal2(A))
=> ( k1_xboole_0 = k1_relat_1(A)
| B = k1_xboole_0
| k8_ordinal2(k4_ordinal3(B,A)) = k15_ordinal2(k8_ordinal2(A),B) ) ) ) ) ).
fof(t53_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( A != k1_xboole_0
=> k3_tarski(k14_ordinal2(B,A)) = k14_ordinal2(B,k3_tarski(A)) ) ) ) ).
fof(t54_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> k15_ordinal2(k14_ordinal2(A,B),C) = k14_ordinal2(k15_ordinal2(A,C),k15_ordinal2(B,C)) ) ) ) ).
fof(t55_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( A != k1_xboole_0
& ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ~ ( B = k14_ordinal2(k15_ordinal2(C,A),D)
& r2_hidden(D,A) ) ) ) ) ) ) ).
fof(t56_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( v3_ordinal1(E)
=> ( ( k14_ordinal2(k15_ordinal2(B,A),C) = k14_ordinal2(k15_ordinal2(D,A),E)
& r2_hidden(C,A)
& r2_hidden(E,A) )
=> ( B = D
& C = E ) ) ) ) ) ) ) ).
fof(t57_ordinal3,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) = k15_ordinal2(D,A) ) ) )
=> k15_ordinal2(B,A) = k8_ordinal2(C) ) ) ) ) ) ) ).
fof(t58_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> k15_ordinal2(k15_ordinal2(A,B),C) = k15_ordinal2(A,k15_ordinal2(B,C)) ) ) ) ).
fof(d6_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( r1_ordinal1(B,A)
=> ( C = k5_ordinal3(A,B)
<=> A = k14_ordinal2(B,C) ) )
& ( ~ r1_ordinal1(B,A)
=> ( C = k5_ordinal3(A,B)
<=> C = k1_xboole_0 ) ) ) ) ) ) ).
fof(d7_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( B != k1_xboole_0
=> ( C = k6_ordinal3(A,B)
<=> ? [D] :
( v3_ordinal1(D)
& A = k14_ordinal2(k15_ordinal2(C,B),D)
& r2_hidden(D,B) ) ) )
& ( B = k1_xboole_0
=> ( C = k6_ordinal3(A,B)
<=> C = k1_xboole_0 ) ) ) ) ) ) ).
fof(d8_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> k7_ordinal3(A,B) = k5_ordinal3(A,k15_ordinal2(k6_ordinal3(A,B),B)) ) ) ).
fof(t59_ordinal3,axiom,
$true ).
fof(t60_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
=> B = k14_ordinal2(A,k5_ordinal3(B,A)) ) ) ) ).
fof(t61_ordinal3,axiom,
$true ).
fof(t62_ordinal3,axiom,
$true ).
fof(t63_ordinal3,axiom,
$true ).
fof(t64_ordinal3,axiom,
$true ).
fof(t65_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> k5_ordinal3(k14_ordinal2(A,B),A) = B ) ) ).
fof(t66_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(A,B)
=> ( ( ~ r1_ordinal1(C,A)
& ~ r2_hidden(C,A) )
| r2_hidden(k5_ordinal3(A,C),k5_ordinal3(B,C)) ) ) ) ) ) ).
fof(t67_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> k5_ordinal3(A,A) = k1_xboole_0 ) ).
fof(t68_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
=> ( k5_ordinal3(B,A) != k1_xboole_0
& r2_hidden(k1_xboole_0,k5_ordinal3(B,A)) ) ) ) ) ).
fof(t69_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ( k5_ordinal3(A,k1_xboole_0) = A
& k5_ordinal3(k1_xboole_0,A) = k1_xboole_0 ) ) ).
fof(t70_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> k5_ordinal3(A,k14_ordinal2(B,C)) = k5_ordinal3(k5_ordinal3(A,B),C) ) ) ) ).
fof(t71_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(A,B)
=> r1_ordinal1(k5_ordinal3(C,B),k5_ordinal3(C,A)) ) ) ) ) ).
fof(t72_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(A,B)
=> r1_ordinal1(k5_ordinal3(A,C),k5_ordinal3(B,C)) ) ) ) ) ).
fof(t73_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(B,k14_ordinal2(C,A))
=> ( A = k1_xboole_0
| r2_hidden(k5_ordinal3(B,C),A) ) ) ) ) ) ).
fof(t74_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(k14_ordinal2(A,B),C)
=> r2_hidden(B,k5_ordinal3(C,A)) ) ) ) ) ).
fof(t75_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> r1_ordinal1(A,k14_ordinal2(B,k5_ordinal3(A,B))) ) ) ).
fof(t76_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> k5_ordinal3(k15_ordinal2(A,B),k15_ordinal2(C,B)) = k15_ordinal2(k5_ordinal3(A,C),B) ) ) ) ).
fof(t77_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> r1_ordinal1(k15_ordinal2(k6_ordinal3(A,B),B),A) ) ) ).
fof(t78_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> A = k14_ordinal2(k15_ordinal2(k6_ordinal3(A,B),B),k7_ordinal3(A,B)) ) ) ).
fof(t79_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ( ( A = k14_ordinal2(k15_ordinal2(B,C),D)
& r2_hidden(D,C) )
=> ( B = k6_ordinal3(A,C)
& D = k7_ordinal3(A,C) ) ) ) ) ) ) ).
fof(t80_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(A,k15_ordinal2(B,C))
=> ( r2_hidden(k6_ordinal3(A,C),B)
& r2_hidden(k7_ordinal3(A,C),C) ) ) ) ) ) ).
fof(t81_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( A != k1_xboole_0
=> k6_ordinal3(k15_ordinal2(B,A),A) = B ) ) ) ).
fof(t82_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> k7_ordinal3(k15_ordinal2(A,B),B) = k1_xboole_0 ) ) ).
fof(t83_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ( k6_ordinal3(k1_xboole_0,A) = k1_xboole_0
& k7_ordinal3(k1_xboole_0,A) = k1_xboole_0
& k7_ordinal3(A,k1_xboole_0) = A ) ) ).
fof(t84_ordinal3,axiom,
! [A] :
( v3_ordinal1(A)
=> ( k6_ordinal3(A,k4_ordinal2) = A
& k7_ordinal3(A,k4_ordinal2) = k1_xboole_0 ) ) ).
fof(dt_k1_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( v1_relat_1(k1_ordinal3(A,B))
& v1_funct_1(k1_ordinal3(A,B))
& v5_ordinal1(k1_ordinal3(A,B))
& v1_ordinal2(k1_ordinal3(A,B)) ) ) ).
fof(dt_k2_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( v1_relat_1(k2_ordinal3(A,B))
& v1_funct_1(k2_ordinal3(A,B))
& v5_ordinal1(k2_ordinal3(A,B))
& v1_ordinal2(k2_ordinal3(A,B)) ) ) ).
fof(dt_k3_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( v1_relat_1(k3_ordinal3(A,B))
& v1_funct_1(k3_ordinal3(A,B))
& v5_ordinal1(k3_ordinal3(A,B))
& v1_ordinal2(k3_ordinal3(A,B)) ) ) ).
fof(dt_k4_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( v1_relat_1(k4_ordinal3(A,B))
& v1_funct_1(k4_ordinal3(A,B))
& v5_ordinal1(k4_ordinal3(A,B))
& v1_ordinal2(k4_ordinal3(A,B)) ) ) ).
fof(dt_k5_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> v3_ordinal1(k5_ordinal3(A,B)) ) ).
fof(dt_k6_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> v3_ordinal1(k6_ordinal3(A,B)) ) ).
fof(dt_k7_ordinal3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> v3_ordinal1(k7_ordinal3(A,B)) ) ).
%------------------------------------------------------------------------------