SET007 Axioms: SET007+15.ax
%------------------------------------------------------------------------------
% File : SET007+15 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Well Ordering 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 : wellord1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 76 ( 11 unt; 0 def)
% Number of atoms : 365 ( 36 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 331 ( 42 ~; 5 |; 119 &)
% ( 16 <=>; 149 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 17 ( 17 usr; 1 con; 0-2 aty)
% Number of variables : 189 ( 187 !; 2 ?)
% 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_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( C = k1_wellord1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( D != B
& r2_hidden(k4_tarski(D,B),A) ) ) ) ) ).
fof(t1_wellord1,axiom,
$true ).
fof(t2_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r2_hidden(A,k3_relat_1(B))
| k1_wellord1(B,A) = k1_xboole_0 ) ) ).
fof(d2_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_wellord1(A)
<=> ! [B] :
~ ( r1_tarski(B,k3_relat_1(A))
& B != k1_xboole_0
& ! [C] :
~ ( r2_hidden(C,B)
& r1_xboole_0(k1_wellord1(A,C),B) ) ) ) ) ).
fof(d3_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_wellord1(A,B)
<=> ! [C] :
~ ( r1_tarski(C,B)
& C != k1_xboole_0
& ! [D] :
~ ( r2_hidden(D,C)
& r1_xboole_0(k1_wellord1(A,D),C) ) ) ) ) ).
fof(t3_wellord1,axiom,
$true ).
fof(t4_wellord1,axiom,
$true ).
fof(t5_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_wellord1(A)
<=> r1_wellord1(A,k3_relat_1(A)) ) ) ).
fof(d4_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_wellord1(A)
<=> ( v1_relat_2(A)
& v8_relat_2(A)
& v4_relat_2(A)
& v6_relat_2(A)
& v1_wellord1(A) ) ) ) ).
fof(d5_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_wellord1(A,B)
<=> ( r1_relat_2(A,B)
& r8_relat_2(A,B)
& r4_relat_2(A,B)
& r6_relat_2(A,B)
& r1_wellord1(A,B) ) ) ) ).
fof(t6_wellord1,axiom,
$true ).
fof(t7_wellord1,axiom,
$true ).
fof(t8_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ( r2_wellord1(A,k3_relat_1(A))
<=> v2_wellord1(A) ) ) ).
fof(t9_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r2_wellord1(B,A)
=> ! [C] :
~ ( r1_tarski(C,A)
& C != k1_xboole_0
& ! [D] :
~ ( r2_hidden(D,C)
& ! [E] :
( r2_hidden(E,C)
=> r2_hidden(k4_tarski(D,E),B) ) ) ) ) ) ).
fof(t10_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_wellord1(A)
=> ! [B] :
~ ( r1_tarski(B,k3_relat_1(A))
& B != k1_xboole_0
& ! [C] :
~ ( r2_hidden(C,B)
& ! [D] :
( r2_hidden(D,B)
=> r2_hidden(k4_tarski(C,D),A) ) ) ) ) ) ).
fof(t11_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ~ ( v2_wellord1(A)
& k3_relat_1(A) != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,k3_relat_1(A))
& ! [C] :
( r2_hidden(C,k3_relat_1(A))
=> r2_hidden(k4_tarski(B,C),A) ) ) ) ) ).
fof(t12_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_wellord1(A)
=> ( k3_relat_1(A) = k1_xboole_0
| ! [B] :
~ ( r2_hidden(B,k3_relat_1(A))
& ? [C] :
( r2_hidden(C,k3_relat_1(A))
& ~ r2_hidden(k4_tarski(C,B),A) )
& ! [C] :
~ ( r2_hidden(C,k3_relat_1(A))
& r2_hidden(k4_tarski(B,C),A)
& ! [D] :
~ ( r2_hidden(D,k3_relat_1(A))
& r2_hidden(k4_tarski(B,D),A)
& D != B
& ~ r2_hidden(k4_tarski(C,D),A) ) ) ) ) ) ) ).
fof(t13_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> r1_tarski(k1_wellord1(B,A),k3_relat_1(B)) ) ).
fof(d6_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] : k2_wellord1(A,B) = k3_xboole_0(A,k2_zfmisc_1(B,B)) ) ).
fof(t14_wellord1,axiom,
$true ).
fof(t15_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r1_tarski(k2_wellord1(B,A),B)
& r1_tarski(k2_wellord1(B,A),k2_zfmisc_1(A,A)) ) ) ).
fof(t16_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( r2_hidden(A,k2_wellord1(C,B))
<=> ( r2_hidden(A,C)
& r2_hidden(A,k2_zfmisc_1(B,B)) ) ) ) ).
fof(t17_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> k2_wellord1(B,A) = k7_relat_1(k8_relat_1(A,B),A) ) ).
fof(t18_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> k2_wellord1(B,A) = k8_relat_1(A,k7_relat_1(B,A)) ) ).
fof(t19_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( r2_hidden(A,k3_relat_1(k2_wellord1(C,B)))
=> ( r2_hidden(A,k3_relat_1(C))
& r2_hidden(A,B) ) ) ) ).
fof(t20_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r1_tarski(k3_relat_1(k2_wellord1(B,A)),k3_relat_1(B))
& r1_tarski(k3_relat_1(k2_wellord1(B,A)),A) ) ) ).
fof(t21_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> r1_tarski(k1_wellord1(k2_wellord1(C,A),B),k1_wellord1(C,B)) ) ).
fof(t22_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v1_relat_2(B)
=> v1_relat_2(k2_wellord1(B,A)) ) ) ).
fof(t23_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v6_relat_2(B)
=> v6_relat_2(k2_wellord1(B,A)) ) ) ).
fof(t24_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v8_relat_2(B)
=> v8_relat_2(k2_wellord1(B,A)) ) ) ).
fof(t25_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v4_relat_2(B)
=> v4_relat_2(k2_wellord1(B,A)) ) ) ).
fof(t26_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> k2_wellord1(k2_wellord1(C,A),B) = k2_wellord1(C,k3_xboole_0(A,B)) ) ).
fof(t27_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> k2_wellord1(k2_wellord1(C,A),B) = k2_wellord1(k2_wellord1(C,B),A) ) ).
fof(t28_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> k2_wellord1(k2_wellord1(B,A),A) = k2_wellord1(B,A) ) ).
fof(t29_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( r1_tarski(A,B)
=> k2_wellord1(k2_wellord1(C,B),A) = k2_wellord1(C,A) ) ) ).
fof(t30_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> k2_wellord1(A,k3_relat_1(A)) = A ) ).
fof(t31_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v1_wellord1(B)
=> v1_wellord1(k2_wellord1(B,A)) ) ) ).
fof(t32_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v2_wellord1(B)
=> v2_wellord1(k2_wellord1(B,A)) ) ) ).
fof(t33_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( v2_wellord1(C)
=> r3_xboole_0(k1_wellord1(C,A),k1_wellord1(C,B)) ) ) ).
fof(t34_wellord1,axiom,
$true ).
fof(t35_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( v2_wellord1(C)
& r2_hidden(A,k3_relat_1(C))
& r2_hidden(B,k1_wellord1(C,A)) )
=> k1_wellord1(k2_wellord1(C,k1_wellord1(C,A)),B) = k1_wellord1(C,B) ) ) ).
fof(t36_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( ( v2_wellord1(B)
& r1_tarski(A,k3_relat_1(B)) )
=> ( ~ ( A != k3_relat_1(B)
& ! [C] :
~ ( r2_hidden(C,k3_relat_1(B))
& A = k1_wellord1(B,C) ) )
<=> ! [C] :
( r2_hidden(C,A)
=> ! [D] :
( r2_hidden(k4_tarski(D,C),B)
=> r2_hidden(D,A) ) ) ) ) ) ).
fof(t37_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( v2_wellord1(C)
& r2_hidden(A,k3_relat_1(C))
& r2_hidden(B,k3_relat_1(C)) )
=> ( r2_hidden(k4_tarski(A,B),C)
<=> r1_tarski(k1_wellord1(C,A),k1_wellord1(C,B)) ) ) ) ).
fof(t38_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( v2_wellord1(C)
& r2_hidden(A,k3_relat_1(C))
& r2_hidden(B,k3_relat_1(C)) )
=> ( r1_tarski(k1_wellord1(C,A),k1_wellord1(C,B))
<=> ( A = B
| r2_hidden(A,k1_wellord1(C,B)) ) ) ) ) ).
fof(t39_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( ( v2_wellord1(B)
& r1_tarski(A,k3_relat_1(B)) )
=> k3_relat_1(k2_wellord1(B,A)) = A ) ) ).
fof(t40_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v2_wellord1(B)
=> k3_relat_1(k2_wellord1(B,k1_wellord1(B,A))) = k1_wellord1(B,A) ) ) ).
fof(t41_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_wellord1(A)
=> ! [B] :
( ! [C] :
( ( r2_hidden(C,k3_relat_1(A))
& r1_tarski(k1_wellord1(A,C),B) )
=> r2_hidden(C,B) )
=> r1_tarski(k3_relat_1(A),B) ) ) ) ).
fof(t42_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( v2_wellord1(C)
& r2_hidden(A,k3_relat_1(C))
& r2_hidden(B,k3_relat_1(C))
& ! [D] :
( r2_hidden(D,k1_wellord1(C,A))
=> ( r2_hidden(k4_tarski(D,B),C)
& D != B ) ) )
=> r2_hidden(k4_tarski(A,B),C) ) ) ).
fof(t43_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_wellord1(A)
& k1_relat_1(B) = k3_relat_1(A)
& r1_tarski(k2_relat_1(B),k3_relat_1(A))
& ! [C,D] :
( r2_hidden(k4_tarski(C,D),A)
=> ( C = D
| ( r2_hidden(k4_tarski(k1_funct_1(B,C),k1_funct_1(B,D)),A)
& k1_funct_1(B,C) != k1_funct_1(B,D) ) ) ) )
=> ! [C] :
( r2_hidden(C,k3_relat_1(A))
=> r2_hidden(k4_tarski(C,k1_funct_1(B,C)),A) ) ) ) ) ).
fof(d7_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r3_wellord1(A,B,C)
<=> ( k1_relat_1(C) = k3_relat_1(A)
& k2_relat_1(C) = k3_relat_1(B)
& v2_funct_1(C)
& ! [D,E] :
( r2_hidden(k4_tarski(D,E),A)
<=> ( r2_hidden(D,k3_relat_1(A))
& r2_hidden(E,k3_relat_1(A))
& r2_hidden(k4_tarski(k1_funct_1(C,D),k1_funct_1(C,E)),B) ) ) ) ) ) ) ) ).
fof(t44_wellord1,axiom,
$true ).
fof(t45_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r3_wellord1(A,B,C)
=> ! [D,E] :
( r2_hidden(k4_tarski(D,E),A)
=> ( D = E
| ( r2_hidden(k4_tarski(k1_funct_1(C,D),k1_funct_1(C,E)),B)
& k1_funct_1(C,D) != k1_funct_1(C,E) ) ) ) ) ) ) ) ).
fof(d8_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r4_wellord1(A,B)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& r3_wellord1(A,B,C) ) ) ) ) ).
fof(t46_wellord1,axiom,
$true ).
fof(t47_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> r3_wellord1(A,A,k6_relat_1(k3_relat_1(A))) ) ).
fof(t48_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> r4_wellord1(A,A) ) ).
fof(t49_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r3_wellord1(A,B,C)
=> r3_wellord1(B,A,k2_funct_1(C)) ) ) ) ) ).
fof(t50_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r4_wellord1(A,B)
=> r4_wellord1(B,A) ) ) ) ).
fof(t51_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r3_wellord1(A,B,D)
& r3_wellord1(B,C,E) )
=> r3_wellord1(A,C,k5_relat_1(D,E)) ) ) ) ) ) ) ).
fof(t52_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> ( ( r4_wellord1(A,B)
& r4_wellord1(B,C) )
=> r4_wellord1(A,C) ) ) ) ) ).
fof(t53_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r3_wellord1(A,B,C)
=> ( ( v1_relat_2(A)
=> v1_relat_2(B) )
& ( v8_relat_2(A)
=> v8_relat_2(B) )
& ( v6_relat_2(A)
=> v6_relat_2(B) )
& ( v4_relat_2(A)
=> v4_relat_2(B) )
& ( v1_wellord1(A)
=> v1_wellord1(B) ) ) ) ) ) ) ).
fof(t54_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( v2_wellord1(A)
& r3_wellord1(A,B,C) )
=> v2_wellord1(B) ) ) ) ) ).
fof(t55_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( v2_wellord1(A)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r3_wellord1(A,B,C)
& r3_wellord1(A,B,D) )
=> C = D ) ) ) ) ) ) ).
fof(d9_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( ( v2_wellord1(A)
& r4_wellord1(A,B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k3_wellord1(A,B)
<=> r3_wellord1(A,B,C) ) ) ) ) ) ).
fof(t56_wellord1,axiom,
$true ).
fof(t57_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_wellord1(A)
=> ! [B] :
~ ( r2_hidden(B,k3_relat_1(A))
& r4_wellord1(A,k2_wellord1(A,k1_wellord1(A,B))) ) ) ) ).
fof(t58_wellord1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ~ ( v2_wellord1(C)
& r2_hidden(A,k3_relat_1(C))
& r2_hidden(B,k3_relat_1(C))
& A != B
& r4_wellord1(k2_wellord1(C,k1_wellord1(C,A)),k2_wellord1(C,k1_wellord1(C,B))) ) ) ).
fof(t59_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( v2_wellord1(B)
& r1_tarski(A,k3_relat_1(B))
& r3_wellord1(B,C,D) )
=> ( r3_wellord1(k2_wellord1(B,A),k2_wellord1(C,k9_relat_1(D,A)),k7_relat_1(D,A))
& r4_wellord1(k2_wellord1(B,A),k2_wellord1(C,k9_relat_1(D,A))) ) ) ) ) ) ).
fof(t60_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( v2_wellord1(A)
& r3_wellord1(A,B,C) )
=> ! [D] :
~ ( r2_hidden(D,k3_relat_1(A))
& ! [E] :
~ ( r2_hidden(E,k3_relat_1(B))
& k9_relat_1(C,k1_wellord1(A,D)) = k1_wellord1(B,E) ) ) ) ) ) ) ).
fof(t61_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( v2_wellord1(A)
& r3_wellord1(A,B,C) )
=> ! [D] :
~ ( r2_hidden(D,k3_relat_1(A))
& ! [E] :
~ ( r2_hidden(E,k3_relat_1(B))
& r4_wellord1(k2_wellord1(A,k1_wellord1(A,D)),k2_wellord1(B,k1_wellord1(B,E))) ) ) ) ) ) ) ).
fof(t62_wellord1,axiom,
! [A,B,C,D] :
( v1_relat_1(D)
=> ! [E] :
( v1_relat_1(E)
=> ( ( v2_wellord1(D)
& v2_wellord1(E)
& r2_hidden(A,k3_relat_1(D))
& r2_hidden(B,k3_relat_1(E))
& r2_hidden(C,k3_relat_1(E))
& r4_wellord1(D,k2_wellord1(E,k1_wellord1(E,B)))
& r4_wellord1(k2_wellord1(D,k1_wellord1(D,A)),k2_wellord1(E,k1_wellord1(E,C))) )
=> ( r1_tarski(k1_wellord1(E,C),k1_wellord1(E,B))
& r2_hidden(k4_tarski(C,B),E) ) ) ) ) ).
fof(t63_wellord1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ~ ( v2_wellord1(A)
& v2_wellord1(B)
& ~ r4_wellord1(A,B)
& ! [C] :
~ ( r2_hidden(C,k3_relat_1(A))
& r4_wellord1(k2_wellord1(A,k1_wellord1(A,C)),B) )
& ! [C] :
~ ( r2_hidden(C,k3_relat_1(B))
& r4_wellord1(A,k2_wellord1(B,k1_wellord1(B,C))) ) ) ) ) ).
fof(t64_wellord1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ~ ( r1_tarski(A,k3_relat_1(B))
& v2_wellord1(B)
& ~ r4_wellord1(B,k2_wellord1(B,A))
& ! [C] :
~ ( r2_hidden(C,k3_relat_1(B))
& r4_wellord1(k2_wellord1(B,k1_wellord1(B,C)),k2_wellord1(B,A)) ) ) ) ).
fof(dt_k1_wellord1,axiom,
$true ).
fof(dt_k2_wellord1,axiom,
! [A,B] :
( v1_relat_1(A)
=> v1_relat_1(k2_wellord1(A,B)) ) ).
fof(dt_k3_wellord1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> ( v1_relat_1(k3_wellord1(A,B))
& v1_funct_1(k3_wellord1(A,B)) ) ) ).
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