SET007 Axioms: SET007+26.ax
%------------------------------------------------------------------------------
% File : SET007+26 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Sequences of 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 : ordinal2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 125 ( 29 unt; 0 def)
% Number of atoms : 834 ( 226 equ)
% Maximal formula atoms : 49 ( 6 avg)
% Number of connectives : 722 ( 13 ~; 30 |; 344 &)
% ( 27 <=>; 308 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-2 aty)
% Number of functors : 95 ( 95 usr; 33 con; 0-2 aty)
% Number of variables : 274 ( 240 !; 34 ?)
% 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_ordinal2,axiom,
( v1_ordinal1(k5_ordinal2)
& v2_ordinal1(k5_ordinal2)
& v3_ordinal1(k5_ordinal2)
& ~ v1_xboole_0(k5_ordinal2) ) ).
fof(rc1_ordinal2,axiom,
? [A] :
( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal1(A) ) ).
fof(rc2_ordinal2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) ) ).
fof(cc1_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( m1_ordinal1(B,A)
=> v1_ordinal2(B) ) ) ).
fof(fc2_ordinal2,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A)
& v3_ordinal1(B) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(A,B))
& v1_ordinal2(k7_relat_1(A,B)) ) ) ).
fof(fc3_ordinal2,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A)
& v3_ordinal1(B) )
=> ( v1_ordinal1(k1_funct_1(A,B))
& v2_ordinal1(k1_funct_1(A,B))
& v3_ordinal1(k1_funct_1(A,B)) ) ) ).
fof(fc4_ordinal2,axiom,
! [A,B] :
( v3_ordinal1(B)
=> ( v1_relat_1(k2_funcop_1(A,B))
& v1_funct_1(k2_funcop_1(A,B))
& v1_ordinal2(k2_funcop_1(A,B)) ) ) ).
fof(t1_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r1_ordinal1(A,B)
<=> r1_ordinal1(k1_ordinal1(A),k1_ordinal1(B)) ) ) ) ).
fof(t2_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k3_tarski(k1_ordinal1(A)) = A ) ).
fof(t3_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> r1_tarski(k1_ordinal1(A),k1_zfmisc_1(A)) ) ).
fof(t4_ordinal2,axiom,
v4_ordinal1(k1_xboole_0) ).
fof(t5_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> r1_ordinal1(k3_tarski(A),A) ) ).
fof(d1_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> k1_ordinal2(A) = k1_funct_1(A,k3_tarski(k1_relat_1(A))) ) ).
fof(t6_ordinal2,axiom,
$true ).
fof(t7_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> ( k1_relat_1(B) = k1_ordinal1(A)
=> k1_ordinal2(B) = k1_funct_1(B,A) ) ) ) ).
fof(d2_ordinal2,axiom,
! [A,B] :
( B = k2_ordinal2(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( r2_hidden(C,A)
& v3_ordinal1(C) ) ) ) ).
fof(d3_ordinal2,axiom,
! [A,B] :
( B = k3_ordinal2(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( r2_hidden(C,A)
& ? [D] :
( v3_ordinal1(D)
& C = D
& v4_ordinal1(D) ) ) ) ) ).
fof(t8_ordinal2,axiom,
$true ).
fof(t9_ordinal2,axiom,
! [A] : r1_tarski(k2_ordinal2(A),A) ).
fof(t10_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k2_ordinal2(A) = A ) ).
fof(t11_ordinal2,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_tarski(k2_ordinal2(A),k2_ordinal2(B)) ) ).
fof(t12_ordinal2,axiom,
$true ).
fof(t13_ordinal2,axiom,
! [A] : r1_tarski(k3_ordinal2(A),A) ).
fof(t14_ordinal2,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_tarski(k3_ordinal2(A),k3_ordinal2(B)) ) ).
fof(t15_ordinal2,axiom,
! [A] : r1_tarski(k3_ordinal2(A),k2_ordinal2(A)) ).
fof(t16_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ? [B] :
( v3_ordinal1(B)
& r2_hidden(A,B)
& v4_ordinal1(B) ) ) ).
fof(t17_ordinal2,axiom,
! [A] :
( ! [B] :
( r2_hidden(B,A)
=> v3_ordinal1(B) )
=> v3_ordinal1(k1_setfam_1(A)) ) ).
fof(d4_ordinal2,axiom,
k4_ordinal2 = k1_ordinal1(k1_xboole_0) ).
fof(d5_ordinal2,axiom,
! [A] :
( A = k5_ordinal2
<=> ( r2_hidden(k1_xboole_0,A)
& v4_ordinal1(A)
& v3_ordinal1(A)
& ! [B] :
( v3_ordinal1(B)
=> ( ( r2_hidden(k1_xboole_0,B)
& v4_ordinal1(B) )
=> r1_tarski(A,B) ) ) ) ) ).
fof(d6_ordinal2,axiom,
! [A] : k6_ordinal2(A) = k1_setfam_1(k2_ordinal2(A)) ).
fof(d7_ordinal2,axiom,
! [A,B] :
( v3_ordinal1(B)
=> ( B = k7_ordinal2(A)
<=> ( r1_tarski(k2_ordinal2(A),B)
& ! [C] :
( v3_ordinal1(C)
=> ( r1_tarski(k2_ordinal2(A),C)
=> r1_ordinal1(B,C) ) ) ) ) ) ).
fof(t18_ordinal2,axiom,
$true ).
fof(t19_ordinal2,axiom,
( r2_hidden(k1_xboole_0,k5_ordinal2)
& v4_ordinal1(k5_ordinal2)
& ! [A] :
( v3_ordinal1(A)
=> ( ( r2_hidden(k1_xboole_0,A)
& v4_ordinal1(A) )
=> r1_ordinal1(k5_ordinal2,A) ) ) ) ).
fof(t20_ordinal2,axiom,
$true ).
fof(t21_ordinal2,axiom,
$true ).
fof(t22_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r2_hidden(A,B)
=> r1_ordinal1(k6_ordinal2(B),A) ) ) ).
fof(t23_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(C,B)
=> r1_ordinal1(A,C) ) )
=> ( k2_ordinal2(B) = k1_xboole_0
| r1_ordinal1(A,k6_ordinal2(B)) ) ) ) ).
fof(t24_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B,C] :
( ( r2_hidden(A,B)
& r1_tarski(B,C) )
=> r1_ordinal1(k6_ordinal2(C),k6_ordinal2(B)) ) ) ).
fof(t25_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r2_hidden(A,B)
=> r2_hidden(k6_ordinal2(B),B) ) ) ).
fof(t26_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k7_ordinal2(A) = A ) ).
fof(t27_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r2_hidden(A,B)
=> r2_hidden(A,k7_ordinal2(B)) ) ) ).
fof(t28_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(C,B)
=> r2_hidden(C,A) ) )
=> r1_ordinal1(k7_ordinal2(B),A) ) ) ).
fof(t29_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
~ ( r2_hidden(A,k7_ordinal2(B))
& ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(C,B)
& r1_ordinal1(A,C) ) ) ) ) ).
fof(t30_ordinal2,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_ordinal1(k7_ordinal2(A),k7_ordinal2(B)) ) ).
fof(t31_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k7_ordinal2(k1_tarski(A)) = k1_ordinal1(A) ) ).
fof(t32_ordinal2,axiom,
! [A] : r1_ordinal1(k6_ordinal2(A),k7_ordinal2(A)) ).
fof(d8_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_ordinal2(A)
<=> ? [B] :
( v3_ordinal1(B)
& r1_tarski(k2_relat_1(A),B) ) ) ) ).
fof(t33_ordinal2,axiom,
$true ).
fof(t34_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( r2_hidden(A,k1_relat_1(B))
=> v3_ordinal1(k1_funct_1(B,A)) ) ) ) ).
fof(d9_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> k8_ordinal2(A) = k7_ordinal2(k2_relat_1(A)) ) ).
fof(d10_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> k9_ordinal2(A) = k6_ordinal2(k2_relat_1(A)) ) ).
fof(t35_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> ( k8_ordinal2(A) = k7_ordinal2(k2_relat_1(A))
& k9_ordinal2(A) = k6_ordinal2(k2_relat_1(A)) ) ) ).
fof(d11_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( B = k10_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& B = k9_ordinal2(C)
& k1_relat_1(C) = k1_relat_1(A)
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,k1_relat_1(A))
=> k1_funct_1(C,D) = k7_ordinal2(k2_relat_1(k7_relat_1(A,k4_xboole_0(k1_relat_1(A),D)))) ) ) ) ) ) ) ).
fof(d12_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( B = k11_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& B = k8_ordinal2(C)
& k1_relat_1(C) = k1_relat_1(A)
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(D,k1_relat_1(A))
=> k1_funct_1(C,D) = k6_ordinal2(k2_relat_1(k7_relat_1(A,k4_xboole_0(k1_relat_1(A),D)))) ) ) ) ) ) ) ).
fof(d13_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( ( A = k1_xboole_0
=> ( r1_ordinal2(A,B)
<=> ? [C] :
( v3_ordinal1(C)
& r2_hidden(C,k1_relat_1(B))
& ! [D] :
( v3_ordinal1(D)
=> ( ( r1_ordinal1(C,D)
& r2_hidden(D,k1_relat_1(B)) )
=> k1_funct_1(B,D) = k1_xboole_0 ) ) ) ) )
& ( A != k1_xboole_0
=> ( r1_ordinal2(A,B)
<=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ~ ( r2_hidden(C,A)
& r2_hidden(A,D)
& ! [E] :
( v3_ordinal1(E)
=> ~ ( r2_hidden(E,k1_relat_1(B))
& ! [F] :
( v3_ordinal1(F)
=> ( ( r1_ordinal1(E,F)
& r2_hidden(F,k1_relat_1(B)) )
=> ( r2_hidden(C,k1_funct_1(B,F))
& r2_hidden(k1_funct_1(B,F),D) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d14_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ( ? [B] :
( v3_ordinal1(B)
& r1_ordinal2(B,A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( B = k12_ordinal2(A)
<=> r1_ordinal2(B,A) ) ) ) ) ).
fof(d15_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> k13_ordinal2(A,B) = k12_ordinal2(k2_ordinal1(B,A)) ) ) ).
fof(d16_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ( v2_ordinal2(A)
<=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( r2_hidden(B,C)
& r2_hidden(C,k1_relat_1(A)) )
=> r2_hidden(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ) ) ).
fof(d17_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ( v3_ordinal2(A)
<=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( r2_hidden(B,k1_relat_1(A))
& v4_ordinal1(B)
& C = k1_funct_1(A,B) )
=> ( B = k1_xboole_0
| r1_ordinal2(C,k2_ordinal1(A,B)) ) ) ) ) ) ) ).
fof(d18_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( C = k14_ordinal2(A,B)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& v1_ordinal2(D)
& C = k1_ordinal2(D)
& k1_relat_1(D) = k1_ordinal1(B)
& k1_funct_1(D,k1_xboole_0) = A
& ! [E] :
( v3_ordinal1(E)
=> ( r2_hidden(k1_ordinal1(E),k1_ordinal1(B))
=> k1_funct_1(D,k1_ordinal1(E)) = k1_ordinal1(k1_funct_1(D,E)) ) )
& ! [E] :
( v3_ordinal1(E)
=> ( ( r2_hidden(E,k1_ordinal1(B))
& v4_ordinal1(E) )
=> ( E = k1_xboole_0
| k1_funct_1(D,E) = k8_ordinal2(k2_ordinal1(D,E)) ) ) ) ) ) ) ) ) ).
fof(d19_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( C = k15_ordinal2(A,B)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& v1_ordinal2(D)
& C = k1_ordinal2(D)
& k1_relat_1(D) = k1_ordinal1(A)
& k1_funct_1(D,k1_xboole_0) = k1_xboole_0
& ! [E] :
( v3_ordinal1(E)
=> ( r2_hidden(k1_ordinal1(E),k1_ordinal1(A))
=> k1_funct_1(D,k1_ordinal1(E)) = k14_ordinal2(k1_funct_1(D,E),B) ) )
& ! [E] :
( v3_ordinal1(E)
=> ( ( r2_hidden(E,k1_ordinal1(A))
& v4_ordinal1(E) )
=> ( E = k1_xboole_0
| k1_funct_1(D,E) = k3_tarski(k8_ordinal2(k2_ordinal1(D,E))) ) ) ) ) ) ) ) ) ).
fof(d20_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( C = k16_ordinal2(A,B)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& v1_ordinal2(D)
& C = k1_ordinal2(D)
& k1_relat_1(D) = k1_ordinal1(B)
& k1_funct_1(D,k1_xboole_0) = k4_ordinal2
& ! [E] :
( v3_ordinal1(E)
=> ( r2_hidden(k1_ordinal1(E),k1_ordinal1(B))
=> k1_funct_1(D,k1_ordinal1(E)) = k15_ordinal2(A,k1_funct_1(D,E)) ) )
& ! [E] :
( v3_ordinal1(E)
=> ( ( r2_hidden(E,k1_ordinal1(B))
& v4_ordinal1(E) )
=> ( E = k1_xboole_0
| k1_funct_1(D,E) = k12_ordinal2(k2_ordinal1(D,E)) ) ) ) ) ) ) ) ) ).
fof(t36_ordinal2,axiom,
$true ).
fof(t37_ordinal2,axiom,
$true ).
fof(t38_ordinal2,axiom,
$true ).
fof(t39_ordinal2,axiom,
$true ).
fof(t40_ordinal2,axiom,
$true ).
fof(t41_ordinal2,axiom,
$true ).
fof(t42_ordinal2,axiom,
$true ).
fof(t43_ordinal2,axiom,
$true ).
fof(t44_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k14_ordinal2(A,k1_xboole_0) = A ) ).
fof(t45_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> k14_ordinal2(A,k1_ordinal1(B)) = k1_ordinal1(k14_ordinal2(A,B)) ) ) ).
fof(t46_ordinal2,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) = k14_ordinal2(B,D) ) ) )
=> k14_ordinal2(B,A) = k8_ordinal2(C) ) ) ) ) ) ) ).
fof(t47_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k14_ordinal2(k1_xboole_0,A) = A ) ).
fof(t48_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k14_ordinal2(A,k4_ordinal2) = k1_ordinal1(A) ) ).
fof(t49_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(A,B)
=> r2_hidden(k14_ordinal2(C,A),k14_ordinal2(C,B)) ) ) ) ) ).
fof(t50_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(A,B)
=> r1_ordinal1(k14_ordinal2(C,A),k14_ordinal2(C,B)) ) ) ) ) ).
fof(t51_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(A,B)
=> r1_ordinal1(k14_ordinal2(A,C),k14_ordinal2(B,C)) ) ) ) ) ).
fof(t52_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k15_ordinal2(k1_xboole_0,A) = k1_xboole_0 ) ).
fof(t53_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> k15_ordinal2(k1_ordinal1(A),B) = k14_ordinal2(k15_ordinal2(A,B),B) ) ) ).
fof(t54_ordinal2,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) = k15_ordinal2(D,B) ) ) )
=> k15_ordinal2(A,B) = k3_tarski(k8_ordinal2(C)) ) ) ) ) ) ) ).
fof(t55_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k15_ordinal2(A,k1_xboole_0) = k1_xboole_0 ) ).
fof(t56_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ( k15_ordinal2(k4_ordinal2,A) = A
& k15_ordinal2(A,k4_ordinal2) = A ) ) ).
fof(t57_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(B,C)
=> ( A = k1_xboole_0
| r2_hidden(k15_ordinal2(B,A),k15_ordinal2(C,A)) ) ) ) ) ) ).
fof(t58_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(A,B)
=> r1_ordinal1(k15_ordinal2(A,C),k15_ordinal2(B,C)) ) ) ) ) ).
fof(t59_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( r1_ordinal1(A,B)
=> r1_ordinal1(k15_ordinal2(C,A),k15_ordinal2(C,B)) ) ) ) ) ).
fof(t60_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> k16_ordinal2(A,k1_xboole_0) = k4_ordinal2 ) ).
fof(t61_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> k16_ordinal2(A,k1_ordinal1(B)) = k15_ordinal2(A,k16_ordinal2(A,B)) ) ) ).
fof(t62_ordinal2,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) ) ) )
=> k16_ordinal2(B,A) = k12_ordinal2(C) ) ) ) ) ) ) ).
fof(t63_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ( k16_ordinal2(A,k4_ordinal2) = A
& k16_ordinal2(k4_ordinal2,A) = k4_ordinal2 ) ) ).
fof(d21_ordinal2,axiom,
! [A] :
( v4_ordinal2(A)
<=> r2_hidden(A,k5_ordinal2) ) ).
fof(t64_ordinal2,axiom,
$true ).
fof(t65_ordinal2,axiom,
! [A] :
( v3_ordinal1(A)
=> ? [B] :
( v3_ordinal1(B)
& ? [C] :
( v3_ordinal1(C)
& v4_ordinal1(B)
& v4_ordinal2(C)
& A = k14_ordinal2(B,C) ) ) ) ).
fof(s1_ordinal2,axiom,
( ( p1_s1_ordinal2(k1_xboole_0)
& ! [A] :
( v3_ordinal1(A)
=> ( p1_s1_ordinal2(A)
=> p1_s1_ordinal2(k1_ordinal1(A)) ) )
& ! [A] :
( v3_ordinal1(A)
=> ( ( v4_ordinal1(A)
& ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(B,A)
=> p1_s1_ordinal2(B) ) ) )
=> ( A = k1_xboole_0
| p1_s1_ordinal2(A) ) ) ) )
=> ! [A] :
( v3_ordinal1(A)
=> p1_s1_ordinal2(A) ) ) ).
fof(s2_ordinal2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& k1_relat_1(A) = f1_s2_ordinal2
& ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(B,f1_s2_ordinal2)
=> k1_funct_1(A,B) = f2_s2_ordinal2(B) ) ) ) ).
fof(s3_ordinal2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A)
& k1_relat_1(A) = f1_s3_ordinal2
& ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(B,f1_s3_ordinal2)
=> k1_funct_1(A,B) = f2_s3_ordinal2(B) ) ) ) ).
fof(s4_ordinal2,axiom,
( ( k1_relat_1(f5_s4_ordinal2) = f1_s4_ordinal2
& ( r2_hidden(k1_xboole_0,f1_s4_ordinal2)
=> k1_funct_1(f5_s4_ordinal2,k1_xboole_0) = f2_s4_ordinal2 )
& ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(k1_ordinal1(A),f1_s4_ordinal2)
=> k1_funct_1(f5_s4_ordinal2,k1_ordinal1(A)) = f3_s4_ordinal2(A,k1_funct_1(f5_s4_ordinal2,A)) ) )
& ! [A] :
( v3_ordinal1(A)
=> ( ( r2_hidden(A,f1_s4_ordinal2)
& v4_ordinal1(A) )
=> ( A = k1_xboole_0
| k1_funct_1(f5_s4_ordinal2,A) = f4_s4_ordinal2(A,k2_ordinal1(f5_s4_ordinal2,A)) ) ) )
& k1_relat_1(f6_s4_ordinal2) = f1_s4_ordinal2
& ( r2_hidden(k1_xboole_0,f1_s4_ordinal2)
=> k1_funct_1(f6_s4_ordinal2,k1_xboole_0) = f2_s4_ordinal2 )
& ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(k1_ordinal1(A),f1_s4_ordinal2)
=> k1_funct_1(f6_s4_ordinal2,k1_ordinal1(A)) = f3_s4_ordinal2(A,k1_funct_1(f6_s4_ordinal2,A)) ) )
& ! [A] :
( v3_ordinal1(A)
=> ( ( r2_hidden(A,f1_s4_ordinal2)
& v4_ordinal1(A) )
=> ( A = k1_xboole_0
| k1_funct_1(f6_s4_ordinal2,A) = f4_s4_ordinal2(A,k2_ordinal1(f6_s4_ordinal2,A)) ) ) ) )
=> f5_s4_ordinal2 = f6_s4_ordinal2 ) ).
fof(s5_ordinal2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& k1_relat_1(A) = f1_s5_ordinal2
& ( r2_hidden(k1_xboole_0,f1_s5_ordinal2)
=> k1_funct_1(A,k1_xboole_0) = f2_s5_ordinal2 )
& ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(k1_ordinal1(B),f1_s5_ordinal2)
=> k1_funct_1(A,k1_ordinal1(B)) = f3_s5_ordinal2(B,k1_funct_1(A,B)) ) )
& ! [B] :
( v3_ordinal1(B)
=> ( ( r2_hidden(B,f1_s5_ordinal2)
& v4_ordinal1(B) )
=> ( B = k1_xboole_0
| k1_funct_1(A,B) = f4_s5_ordinal2(B,k2_ordinal1(A,B)) ) ) ) ) ).
fof(s6_ordinal2,axiom,
( ( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( B = f2_s6_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = f4_s6_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = f5_s6_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f6_s6_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
& k1_relat_1(f1_s6_ordinal2) = f3_s6_ordinal2
& ( r2_hidden(k1_xboole_0,f3_s6_ordinal2)
=> k1_funct_1(f1_s6_ordinal2,k1_xboole_0) = f4_s6_ordinal2 )
& ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(k1_ordinal1(A),f3_s6_ordinal2)
=> k1_funct_1(f1_s6_ordinal2,k1_ordinal1(A)) = f5_s6_ordinal2(A,k1_funct_1(f1_s6_ordinal2,A)) ) )
& ! [A] :
( v3_ordinal1(A)
=> ( ( r2_hidden(A,f3_s6_ordinal2)
& v4_ordinal1(A) )
=> ( A = k1_xboole_0
| k1_funct_1(f1_s6_ordinal2,A) = f6_s6_ordinal2(A,k2_ordinal1(f1_s6_ordinal2,A)) ) ) ) )
=> ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(A,k1_relat_1(f1_s6_ordinal2))
=> k1_funct_1(f1_s6_ordinal2,A) = f2_s6_ordinal2(A) ) ) ) ).
fof(s7_ordinal2,axiom,
( ? [A,B] :
( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& A = k1_ordinal2(B)
& k1_relat_1(B) = k1_ordinal1(f1_s7_ordinal2)
& k1_funct_1(B,k1_xboole_0) = f2_s7_ordinal2
& ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(k1_ordinal1(C),k1_ordinal1(f1_s7_ordinal2))
=> k1_funct_1(B,k1_ordinal1(C)) = f3_s7_ordinal2(C,k1_funct_1(B,C)) ) )
& ! [C] :
( v3_ordinal1(C)
=> ( ( r2_hidden(C,k1_ordinal1(f1_s7_ordinal2))
& v4_ordinal1(C) )
=> ( C = k1_xboole_0
| k1_funct_1(B,C) = f4_s7_ordinal2(C,k2_ordinal1(B,C)) ) ) ) )
& ! [A,B] :
~ ( ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& A = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(f1_s7_ordinal2)
& k1_funct_1(C,k1_xboole_0) = f2_s7_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s7_ordinal2))
=> k1_funct_1(C,k1_ordinal1(D)) = f3_s7_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(f1_s7_ordinal2))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f4_s7_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
& ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(f1_s7_ordinal2)
& k1_funct_1(C,k1_xboole_0) = f2_s7_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s7_ordinal2))
=> k1_funct_1(C,k1_ordinal1(D)) = f3_s7_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(f1_s7_ordinal2))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f4_s7_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
& A != B ) ) ).
fof(s8_ordinal2,axiom,
( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( B = f1_s8_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = f2_s8_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = f3_s8_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f4_s8_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
=> f1_s8_ordinal2(k1_xboole_0) = f2_s8_ordinal2 ) ).
fof(s9_ordinal2,axiom,
( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( B = f4_s9_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = f1_s9_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = f2_s9_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f3_s9_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
=> ! [A] :
( v3_ordinal1(A)
=> f4_s9_ordinal2(k1_ordinal1(A)) = f2_s9_ordinal2(A,f4_s9_ordinal2(A)) ) ) ).
fof(s10_ordinal2,axiom,
( ( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( B = f3_s10_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = f4_s10_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = f5_s10_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f6_s10_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
& f2_s10_ordinal2 != k1_xboole_0
& v4_ordinal1(f2_s10_ordinal2)
& k1_relat_1(f1_s10_ordinal2) = f2_s10_ordinal2
& ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(A,f2_s10_ordinal2)
=> k1_funct_1(f1_s10_ordinal2,A) = f3_s10_ordinal2(A) ) ) )
=> f3_s10_ordinal2(f2_s10_ordinal2) = f6_s10_ordinal2(f2_s10_ordinal2,f1_s10_ordinal2) ) ).
fof(s11_ordinal2,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A)
& k1_relat_1(A) = f1_s11_ordinal2
& ( r2_hidden(k1_xboole_0,f1_s11_ordinal2)
=> k1_funct_1(A,k1_xboole_0) = f2_s11_ordinal2 )
& ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(k1_ordinal1(B),f1_s11_ordinal2)
=> k1_funct_1(A,k1_ordinal1(B)) = f3_s11_ordinal2(B,k1_funct_1(A,B)) ) )
& ! [B] :
( v3_ordinal1(B)
=> ( ( r2_hidden(B,f1_s11_ordinal2)
& v4_ordinal1(B) )
=> ( B = k1_xboole_0
| k1_funct_1(A,B) = f4_s11_ordinal2(B,k2_ordinal1(A,B)) ) ) ) ) ).
fof(s12_ordinal2,axiom,
( ( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( B = f2_s12_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = f4_s12_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = f5_s12_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f6_s12_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
& k1_relat_1(f1_s12_ordinal2) = f3_s12_ordinal2
& ( r2_hidden(k1_xboole_0,f3_s12_ordinal2)
=> k1_funct_1(f1_s12_ordinal2,k1_xboole_0) = f4_s12_ordinal2 )
& ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(k1_ordinal1(A),f3_s12_ordinal2)
=> k1_funct_1(f1_s12_ordinal2,k1_ordinal1(A)) = f5_s12_ordinal2(A,k1_funct_1(f1_s12_ordinal2,A)) ) )
& ! [A] :
( v3_ordinal1(A)
=> ( ( r2_hidden(A,f3_s12_ordinal2)
& v4_ordinal1(A) )
=> ( A = k1_xboole_0
| k1_funct_1(f1_s12_ordinal2,A) = f6_s12_ordinal2(A,k2_ordinal1(f1_s12_ordinal2,A)) ) ) ) )
=> ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(A,k1_relat_1(f1_s12_ordinal2))
=> k1_funct_1(f1_s12_ordinal2,A) = f2_s12_ordinal2(A) ) ) ) ).
fof(s13_ordinal2,axiom,
( ? [A] :
( v3_ordinal1(A)
& ? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B)
& A = k1_ordinal2(B)
& k1_relat_1(B) = k1_ordinal1(f1_s13_ordinal2)
& k1_funct_1(B,k1_xboole_0) = f2_s13_ordinal2
& ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(k1_ordinal1(C),k1_ordinal1(f1_s13_ordinal2))
=> k1_funct_1(B,k1_ordinal1(C)) = f3_s13_ordinal2(C,k1_funct_1(B,C)) ) )
& ! [C] :
( v3_ordinal1(C)
=> ( ( r2_hidden(C,k1_ordinal1(f1_s13_ordinal2))
& v4_ordinal1(C) )
=> ( C = k1_xboole_0
| k1_funct_1(B,C) = f4_s13_ordinal2(C,k2_ordinal1(B,C)) ) ) ) ) )
& ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ~ ( ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& A = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(f1_s13_ordinal2)
& k1_funct_1(C,k1_xboole_0) = f2_s13_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s13_ordinal2))
=> k1_funct_1(C,k1_ordinal1(D)) = f3_s13_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(f1_s13_ordinal2))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f4_s13_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
& ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(f1_s13_ordinal2)
& k1_funct_1(C,k1_xboole_0) = f2_s13_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s13_ordinal2))
=> k1_funct_1(C,k1_ordinal1(D)) = f3_s13_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(f1_s13_ordinal2))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f4_s13_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
& A != B ) ) ) ) ).
fof(s14_ordinal2,axiom,
( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( B = f1_s14_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = f2_s14_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = f3_s14_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f4_s14_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
=> f1_s14_ordinal2(k1_xboole_0) = f2_s14_ordinal2 ) ).
fof(s15_ordinal2,axiom,
( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( B = f4_s15_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = f1_s15_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = f2_s15_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f3_s15_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
=> ! [A] :
( v3_ordinal1(A)
=> f4_s15_ordinal2(k1_ordinal1(A)) = f2_s15_ordinal2(A,f4_s15_ordinal2(A)) ) ) ).
fof(s16_ordinal2,axiom,
( ( ! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( B = f3_s16_ordinal2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = f4_s16_ordinal2
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = f5_s16_ordinal2(D,k1_funct_1(C,D)) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = f6_s16_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
& f2_s16_ordinal2 != k1_xboole_0
& v4_ordinal1(f2_s16_ordinal2)
& k1_relat_1(f1_s16_ordinal2) = f2_s16_ordinal2
& ! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(A,f2_s16_ordinal2)
=> k1_funct_1(f1_s16_ordinal2,A) = f3_s16_ordinal2(A) ) ) )
=> f3_s16_ordinal2(f2_s16_ordinal2) = f6_s16_ordinal2(f2_s16_ordinal2,f1_s16_ordinal2) ) ).
fof(dt_k1_ordinal2,axiom,
$true ).
fof(dt_k2_ordinal2,axiom,
$true ).
fof(dt_k3_ordinal2,axiom,
$true ).
fof(dt_k4_ordinal2,axiom,
( v3_ordinal1(k4_ordinal2)
& ~ v1_xboole_0(k4_ordinal2) ) ).
fof(dt_k5_ordinal2,axiom,
$true ).
fof(dt_k6_ordinal2,axiom,
! [A] : v3_ordinal1(k6_ordinal2(A)) ).
fof(dt_k7_ordinal2,axiom,
! [A] : v3_ordinal1(k7_ordinal2(A)) ).
fof(dt_k8_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> v3_ordinal1(k8_ordinal2(A)) ) ).
fof(dt_k9_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> v3_ordinal1(k9_ordinal2(A)) ) ).
fof(dt_k10_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> v3_ordinal1(k10_ordinal2(A)) ) ).
fof(dt_k11_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> v3_ordinal1(k11_ordinal2(A)) ) ).
fof(dt_k12_ordinal2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> v3_ordinal1(k12_ordinal2(A)) ) ).
fof(dt_k13_ordinal2,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> v3_ordinal1(k13_ordinal2(A,B)) ) ).
fof(dt_k14_ordinal2,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> v3_ordinal1(k14_ordinal2(A,B)) ) ).
fof(dt_k15_ordinal2,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> v3_ordinal1(k15_ordinal2(A,B)) ) ).
fof(dt_k16_ordinal2,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> v3_ordinal1(k16_ordinal2(A,B)) ) ).
%------------------------------------------------------------------------------