SET007 Axioms: SET007+28.ax
%------------------------------------------------------------------------------
% File : SET007+28 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Arithmetic of Non-Negative Rational 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 : arytm_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 161 ( 13 unt; 0 def)
% Number of atoms : 859 ( 166 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 828 ( 130 ~; 6 |; 336 &)
% ( 22 <=>; 334 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-2 aty)
% Number of functors : 32 ( 32 usr; 11 con; 0-2 aty)
% Number of variables : 332 ( 317 !; 15 ?)
% 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(cc1_arytm_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_ordinal1(B)
& v2_ordinal1(B)
& v3_ordinal1(B) ) ) ) ).
fof(cc2_arytm_3,axiom,
! [A] :
( ( v1_xboole_0(A)
& v3_ordinal1(A) )
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A) ) ) ).
fof(fc1_arytm_3,axiom,
( ~ v1_xboole_0(k4_ordinal2)
& v1_ordinal1(k4_ordinal2)
& v2_ordinal1(k4_ordinal2)
& v3_ordinal1(k4_ordinal2)
& v4_ordinal2(k4_ordinal2) ) ).
fof(cc3_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k5_ordinal2)
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A) ) ) ).
fof(rc1_arytm_3,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A) ) ).
fof(fc2_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( ~ v1_xboole_0(k1_ordinal1(A))
& v1_ordinal1(k1_ordinal1(A))
& v2_ordinal1(k1_ordinal1(A))
& v3_ordinal1(k1_ordinal1(A))
& v4_ordinal2(k1_ordinal1(A)) ) ) ).
fof(fc3_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( v1_ordinal1(k14_ordinal2(A,B))
& v2_ordinal1(k14_ordinal2(A,B))
& v3_ordinal1(k14_ordinal2(A,B))
& v4_ordinal2(k14_ordinal2(A,B)) ) ) ).
fof(fc4_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( v1_ordinal1(k5_ordinal3(A,B))
& v2_ordinal1(k5_ordinal3(A,B))
& v3_ordinal1(k5_ordinal3(A,B))
& v4_ordinal2(k5_ordinal3(A,B)) ) ) ).
fof(fc5_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( v1_ordinal1(k15_ordinal2(A,B))
& v2_ordinal1(k15_ordinal2(A,B))
& v3_ordinal1(k15_ordinal2(A,B))
& v4_ordinal2(k15_ordinal2(A,B)) ) ) ).
fof(fc6_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( v1_ordinal1(k6_ordinal3(A,B))
& v2_ordinal1(k6_ordinal3(A,B))
& v3_ordinal1(k6_ordinal3(A,B))
& v4_ordinal2(k6_ordinal3(A,B)) ) ) ).
fof(fc7_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( v1_ordinal1(k7_ordinal3(A,B))
& v2_ordinal1(k7_ordinal3(A,B))
& v3_ordinal1(k7_ordinal3(A,B))
& v4_ordinal2(k7_ordinal3(A,B)) ) ) ).
fof(fc8_arytm_3,axiom,
~ v1_xboole_0(k6_arytm_3) ).
fof(rc2_arytm_3,axiom,
? [A] :
( m1_subset_1(A,k6_arytm_3)
& ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A) ) ).
fof(cc4_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ( v3_ordinal1(A)
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A) ) ) ) ).
fof(rc3_arytm_3,axiom,
? [A] :
( m1_subset_1(A,k6_arytm_3)
& v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A) ) ).
fof(t1_arytm_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( v4_ordinal2(k14_ordinal2(A,B))
=> ( r2_hidden(A,k5_ordinal2)
& r2_hidden(B,k5_ordinal2) ) ) ) ) ).
fof(t2_arytm_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( v4_ordinal2(k15_ordinal2(A,B))
=> ( v1_xboole_0(k15_ordinal2(A,B))
| ( r2_hidden(A,k5_ordinal2)
& r2_hidden(B,k5_ordinal2) ) ) ) ) ) ).
fof(t3_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> k14_ordinal2(A,B) = k14_ordinal2(B,A) ) ) ).
fof(t4_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> k15_ordinal2(A,B) = k15_ordinal2(B,A) ) ) ).
fof(d1_arytm_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r1_arytm_3(A,B)
<=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( v3_ordinal1(E)
=> ( ( A = k15_ordinal2(C,D)
& B = k15_ordinal2(C,E) )
=> C = k4_ordinal2 ) ) ) ) ) ) ) ).
fof(t5_arytm_3,axiom,
~ r1_arytm_3(k1_xboole_0,k1_xboole_0) ).
fof(t6_arytm_3,axiom,
! [A] :
( v3_ordinal1(A)
=> r1_arytm_3(k4_ordinal2,A) ) ).
fof(t7_arytm_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ( r1_arytm_3(k1_xboole_0,A)
=> A = k4_ordinal2 ) ) ).
fof(t8_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ~ ( ~ ( A = k1_xboole_0
& B = k1_xboole_0 )
& ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> ! [D] :
( ( v3_ordinal1(D)
& v4_ordinal2(D) )
=> ! [E] :
( ( v3_ordinal1(E)
& v4_ordinal2(E) )
=> ~ ( r1_arytm_3(D,E)
& A = k2_arytm_3(C,D)
& B = k2_arytm_3(C,E) ) ) ) ) ) ) ) ).
fof(d2_arytm_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_arytm_3(A,B)
<=> ? [C] :
( v3_ordinal1(C)
& B = k15_ordinal2(A,C) ) ) ) ) ).
fof(t9_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( r2_arytm_3(A,B)
<=> ? [C] :
( v3_ordinal1(C)
& v4_ordinal2(C)
& B = k2_arytm_3(A,C) ) ) ) ) ).
fof(t10_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( r2_hidden(k1_xboole_0,A)
=> r2_hidden(k7_ordinal3(B,A),A) ) ) ) ).
fof(t11_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( r2_arytm_3(B,A)
<=> A = k2_arytm_3(B,k6_ordinal3(A,B)) ) ) ) ).
fof(t12_arytm_3,axiom,
$true ).
fof(t13_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( ( r2_arytm_3(A,B)
& r2_arytm_3(B,A) )
=> A = B ) ) ) ).
fof(t14_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( r2_arytm_3(A,k1_xboole_0)
& r2_arytm_3(k4_ordinal2,A) ) ) ).
fof(t15_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( ( r2_hidden(k1_xboole_0,B)
& r2_arytm_3(A,B) )
=> r1_ordinal1(A,B) ) ) ) ).
fof(t16_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> ( ( r2_arytm_3(A,B)
& r2_arytm_3(A,k1_arytm_3(B,C)) )
=> r2_arytm_3(A,C) ) ) ) ) ).
fof(d3_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( m1_subset_1(C,k5_ordinal2)
=> ( C = k3_arytm_3(A,B)
<=> ( r2_arytm_3(A,C)
& r2_arytm_3(B,C)
& ! [D] :
( ( v3_ordinal1(D)
& v4_ordinal2(D) )
=> ( ( r2_arytm_3(A,D)
& r2_arytm_3(B,D) )
=> r2_arytm_3(C,D) ) ) ) ) ) ) ) ).
fof(t17_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> r2_arytm_3(k3_arytm_3(A,B),k2_arytm_3(A,B)) ) ) ).
fof(t18_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( A != k1_xboole_0
=> r2_arytm_3(k6_ordinal3(k2_arytm_3(B,A),k3_arytm_3(B,A)),B) ) ) ) ).
fof(d4_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( m1_subset_1(C,k5_ordinal2)
=> ( C = k4_arytm_3(A,B)
<=> ( r2_arytm_3(C,A)
& r2_arytm_3(C,B)
& ! [D] :
( ( v3_ordinal1(D)
& v4_ordinal2(D) )
=> ( ( r2_arytm_3(D,A)
& r2_arytm_3(D,B) )
=> r2_arytm_3(D,C) ) ) ) ) ) ) ) ).
fof(t19_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( k4_arytm_3(A,k1_xboole_0) = A
& k3_arytm_3(A,k1_xboole_0) = k1_xboole_0 ) ) ).
fof(t20_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( k4_arytm_3(A,B) = k1_xboole_0
=> A = k1_xboole_0 ) ) ) ).
fof(t21_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( k4_arytm_3(A,A) = A
& k3_arytm_3(A,A) = A ) ) ).
fof(t22_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> k4_arytm_3(k2_arytm_3(A,B),k2_arytm_3(C,B)) = k2_arytm_3(k4_arytm_3(A,C),B) ) ) ) ).
fof(t23_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( A != k1_xboole_0
=> ( k4_arytm_3(B,A) != k1_xboole_0
& k6_ordinal3(A,k4_arytm_3(B,A)) != k1_xboole_0 ) ) ) ) ).
fof(t24_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( ~ ( A = k1_xboole_0
& B = k1_xboole_0 )
=> r1_arytm_3(k6_ordinal3(A,k4_arytm_3(A,B)),k6_ordinal3(B,k4_arytm_3(A,B))) ) ) ) ).
fof(t25_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( r1_arytm_3(A,B)
<=> k4_arytm_3(A,B) = k4_ordinal2 ) ) ) ).
fof(d5_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> k5_arytm_3(A,B) = k6_ordinal3(A,k4_arytm_3(A,B)) ) ) ).
fof(t26_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> k2_arytm_3(k5_arytm_3(A,B),k4_arytm_3(A,B)) = A ) ) ).
fof(t27_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( ~ ( A = k1_xboole_0
& B = k1_xboole_0 )
=> r1_arytm_3(k5_arytm_3(A,B),k5_arytm_3(B,A)) ) ) ) ).
fof(t28_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( r1_arytm_3(A,B)
=> k5_arytm_3(A,B) = A ) ) ) ).
fof(t29_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( k5_arytm_3(A,k4_ordinal2) = A
& k5_arytm_3(k4_ordinal2,A) = k4_ordinal2 ) ) ).
fof(t30_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ~ ( A != k1_xboole_0
& k5_arytm_3(A,B) = k1_xboole_0 ) ) ) ).
fof(t31_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( k5_arytm_3(k1_xboole_0,A) = k1_xboole_0
& ( A != k1_xboole_0
=> k5_arytm_3(A,k1_xboole_0) = k4_ordinal2 ) ) ) ).
fof(t32_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( A != k1_xboole_0
=> k5_arytm_3(A,A) = k4_ordinal2 ) ) ).
fof(t33_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> ( A != k1_xboole_0
=> k5_arytm_3(k2_arytm_3(B,A),k2_arytm_3(C,A)) = k5_arytm_3(B,C) ) ) ) ) ).
fof(t34_arytm_3,axiom,
r1_tarski(k5_ordinal2,k6_arytm_3) ).
fof(t35_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ~ ( ~ r2_hidden(A,k5_ordinal2)
& ! [B] :
( m1_subset_1(B,k5_ordinal2)
=> ! [C] :
( m1_subset_1(C,k5_ordinal2)
=> ~ ( A = k4_tarski(B,C)
& r1_arytm_3(B,C)
& C != k1_xboole_0
& C != k4_ordinal2 ) ) ) ) ) ).
fof(t36_arytm_3,axiom,
! [A,B] : ~ v3_ordinal1(k4_tarski(A,B)) ).
fof(t37_arytm_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ( r2_hidden(A,k6_arytm_3)
=> r2_hidden(A,k5_ordinal2) ) ) ).
fof(t38_arytm_3,axiom,
! [A,B] : ~ r2_hidden(k4_tarski(A,B),k5_ordinal2) ).
fof(t39_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k5_ordinal2)
=> ! [B] :
( m1_subset_1(B,k5_ordinal2)
=> ( r2_hidden(k4_tarski(A,B),k6_arytm_3)
<=> ( r1_arytm_3(A,B)
& B != k1_xboole_0
& B != k4_ordinal2 ) ) ) ) ).
fof(d7_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k5_ordinal2)
=> ( ( r2_hidden(A,k5_ordinal2)
=> ( B = k7_arytm_3(A)
<=> B = A ) )
& ( ~ r2_hidden(A,k5_ordinal2)
=> ( B = k7_arytm_3(A)
<=> ? [C] :
( v3_ordinal1(C)
& v4_ordinal2(C)
& A = k4_tarski(B,C) ) ) ) ) ) ) ).
fof(d8_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k5_ordinal2)
=> ( ( r2_hidden(A,k5_ordinal2)
=> ( B = k8_arytm_3(A)
<=> B = k4_ordinal2 ) )
& ( ~ r2_hidden(A,k5_ordinal2)
=> ( B = k8_arytm_3(A)
<=> ? [C] :
( v3_ordinal1(C)
& v4_ordinal2(C)
& A = k4_tarski(C,B) ) ) ) ) ) ) ).
fof(t40_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> r1_arytm_3(k7_arytm_3(A),k8_arytm_3(A)) ) ).
fof(t41_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> k8_arytm_3(A) != k1_xboole_0 ) ).
fof(t42_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ( ~ r2_hidden(A,k5_ordinal2)
=> ( A = k4_tarski(k7_arytm_3(A),k8_arytm_3(A))
& k8_arytm_3(A) != k4_ordinal2 ) ) ) ).
fof(t43_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ( ~ ( A != k1_xboole_0
& k7_arytm_3(A) = k1_xboole_0 )
& ~ ( k7_arytm_3(A) != k1_xboole_0
& A = k1_xboole_0 ) ) ) ).
fof(t44_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ( r2_hidden(A,k5_ordinal2)
<=> k8_arytm_3(A) = k4_ordinal2 ) ) ).
fof(d9_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( ( B = k1_xboole_0
=> k9_arytm_3(A,B) = k1_xboole_0 )
& ( k5_arytm_3(B,A) = k4_ordinal2
=> k9_arytm_3(A,B) = k5_arytm_3(A,B) )
& ~ ( B != k1_xboole_0
& k5_arytm_3(B,A) != k4_ordinal2
& k9_arytm_3(A,B) != k4_tarski(k5_arytm_3(A,B),k5_arytm_3(B,A)) ) ) ) ) ).
fof(t45_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> k9_arytm_3(k7_arytm_3(A),k8_arytm_3(A)) = A ) ).
fof(t46_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( k9_arytm_3(k1_xboole_0,A) = k1_xboole_0
& k9_arytm_3(B,k4_ordinal2) = B ) ) ) ).
fof(t47_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( A != k1_xboole_0
=> k9_arytm_3(A,A) = k4_ordinal2 ) ) ).
fof(t48_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ( A != k1_xboole_0
=> ( k7_arytm_3(k9_arytm_3(B,A)) = k5_arytm_3(B,A)
& k8_arytm_3(k9_arytm_3(B,A)) = k5_arytm_3(A,B) ) ) ) ) ).
fof(t49_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k5_ordinal2)
=> ! [B] :
( m1_subset_1(B,k5_ordinal2)
=> ( r1_arytm_3(A,B)
=> ( B = k1_xboole_0
| ( k7_arytm_3(k9_arytm_3(A,B)) = A
& k8_arytm_3(k9_arytm_3(A,B)) = B ) ) ) ) ) ).
fof(t50_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> ( A != k1_xboole_0
=> k9_arytm_3(k2_arytm_3(B,A),k2_arytm_3(C,A)) = k9_arytm_3(B,C) ) ) ) ) ).
fof(t51_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> ! [D] :
( ( v3_ordinal1(D)
& v4_ordinal2(D) )
=> ~ ( B != k1_xboole_0
& A != k1_xboole_0
& ~ ( k9_arytm_3(C,B) = k9_arytm_3(D,A)
<=> k2_arytm_3(C,A) = k2_arytm_3(B,D) ) ) ) ) ) ) ).
fof(d10_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> k10_arytm_3(A,B) = k9_arytm_3(k1_arytm_3(k2_arytm_3(k7_arytm_3(A),k8_arytm_3(B)),k2_arytm_3(k7_arytm_3(B),k8_arytm_3(A))),k2_arytm_3(k8_arytm_3(A),k8_arytm_3(B))) ) ) ).
fof(d11_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> k11_arytm_3(A,B) = k9_arytm_3(k2_arytm_3(k7_arytm_3(A),k7_arytm_3(B)),k2_arytm_3(k8_arytm_3(A),k8_arytm_3(B))) ) ) ).
fof(t52_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> ! [D] :
( ( v3_ordinal1(D)
& v4_ordinal2(D) )
=> ~ ( B != k1_xboole_0
& A != k1_xboole_0
& k10_arytm_3(k9_arytm_3(C,B),k9_arytm_3(D,A)) != k9_arytm_3(k1_arytm_3(k2_arytm_3(C,A),k2_arytm_3(B,D)),k2_arytm_3(B,A)) ) ) ) ) ) ).
fof(t53_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> ( A != k1_xboole_0
=> k10_arytm_3(k9_arytm_3(B,A),k9_arytm_3(C,A)) = k9_arytm_3(k1_arytm_3(B,C),A) ) ) ) ) ).
fof(t54_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> k11_arytm_3(A,k12_arytm_3) = k12_arytm_3 ) ).
fof(t55_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal2(C) )
=> ! [D] :
( ( v3_ordinal1(D)
& v4_ordinal2(D) )
=> k11_arytm_3(k9_arytm_3(B,C),k9_arytm_3(D,A)) = k9_arytm_3(k2_arytm_3(B,D),k2_arytm_3(C,A)) ) ) ) ) ).
fof(t56_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> k10_arytm_3(A,k12_arytm_3) = A ) ).
fof(t57_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> k10_arytm_3(k10_arytm_3(A,B),C) = k10_arytm_3(A,k10_arytm_3(B,C)) ) ) ) ).
fof(t58_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> k11_arytm_3(k11_arytm_3(A,B),C) = k11_arytm_3(A,k11_arytm_3(B,C)) ) ) ) ).
fof(t59_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> k11_arytm_3(A,k13_arytm_3) = A ) ).
fof(t60_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ~ ( A != k12_arytm_3
& ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> k11_arytm_3(A,B) != k13_arytm_3 ) ) ) ).
fof(t61_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ~ ( A != k12_arytm_3
& ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> B != k11_arytm_3(A,C) ) ) ) ) ).
fof(t62_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( k11_arytm_3(A,B) = k11_arytm_3(A,C)
=> ( A = k12_arytm_3
| B = C ) ) ) ) ) ).
fof(t63_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> k11_arytm_3(A,k10_arytm_3(B,C)) = k10_arytm_3(k11_arytm_3(A,B),k11_arytm_3(A,C)) ) ) ) ).
fof(t64_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& m1_subset_1(A,k6_arytm_3) )
=> ! [B] :
( ( v3_ordinal1(B)
& m1_subset_1(B,k6_arytm_3) )
=> k10_arytm_3(A,B) = k1_arytm_3(A,B) ) ) ).
fof(t65_arytm_3,axiom,
! [A] :
( ( v3_ordinal1(A)
& m1_subset_1(A,k6_arytm_3) )
=> ! [B] :
( ( v3_ordinal1(B)
& m1_subset_1(B,k6_arytm_3) )
=> k11_arytm_3(A,B) = k2_arytm_3(A,B) ) ) ).
fof(t66_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ? [B] :
( m1_subset_1(B,k6_arytm_3)
& A = k10_arytm_3(B,B) ) ) ).
fof(d12_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ( r3_arytm_3(A,B)
<=> ? [C] :
( m1_subset_1(C,k6_arytm_3)
& B = k10_arytm_3(A,C) ) ) ) ) ).
fof(t67_arytm_3,axiom,
$true ).
fof(t68_arytm_3,axiom,
! [A] : ~ r2_hidden(k4_tarski(k12_arytm_3,A),k6_arytm_3) ).
fof(t69_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( k10_arytm_3(A,B) = k10_arytm_3(C,B)
=> A = C ) ) ) ) ).
fof(t70_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ( k10_arytm_3(A,B) = k12_arytm_3
=> A = k12_arytm_3 ) ) ) ).
fof(t71_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> r3_arytm_3(k12_arytm_3,A) ) ).
fof(t72_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ( r3_arytm_3(A,k12_arytm_3)
=> A = k12_arytm_3 ) ) ).
fof(t73_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ( ( r3_arytm_3(A,B)
& r3_arytm_3(B,A) )
=> A = B ) ) ) ).
fof(t74_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( ( r3_arytm_3(A,B)
& r3_arytm_3(B,C) )
=> r3_arytm_3(A,C) ) ) ) ) ).
fof(t75_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ( ~ r3_arytm_3(B,A)
<=> ( r3_arytm_3(A,B)
& A != B ) ) ) ) ).
fof(t76_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( ( ( ~ r3_arytm_3(B,A)
& r3_arytm_3(B,C) )
| ( r3_arytm_3(A,B)
& ~ r3_arytm_3(C,B) ) )
& r3_arytm_3(C,A) ) ) ) ) ).
fof(t77_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( ~ r3_arytm_3(B,A)
& ~ r3_arytm_3(C,B)
& r3_arytm_3(C,A) ) ) ) ) ).
fof(t78_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ( ( r2_hidden(A,k5_ordinal2)
& r2_hidden(k10_arytm_3(A,B),k5_ordinal2) )
=> r2_hidden(B,k5_ordinal2) ) ) ) ).
fof(t79_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( ( v3_ordinal1(B)
& m1_subset_1(B,k6_arytm_3) )
=> ~ ( ~ r3_arytm_3(A,B)
& ~ r3_arytm_3(k10_arytm_3(B,k13_arytm_3),A)
& r2_hidden(A,k5_ordinal2) ) ) ) ).
fof(t80_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ~ ( A != k12_arytm_3
& ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ~ ( ~ r3_arytm_3(A,B)
& ~ r2_hidden(B,k5_ordinal2) ) ) ) ) ).
fof(t82_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k6_arytm_3))
=> ~ ( ? [B] :
( m1_subset_1(B,k6_arytm_3)
& r2_hidden(B,A)
& B != k12_arytm_3 )
& ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( ( r2_hidden(B,A)
& r3_arytm_3(C,B) )
=> r2_hidden(C,A) ) ) )
& ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ~ ( r2_hidden(B,A)
& r2_hidden(C,A)
& r2_hidden(D,A)
& B != C
& C != D
& D != B ) ) ) ) ) ) ).
fof(t83_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( r3_arytm_3(k10_arytm_3(A,B),k10_arytm_3(C,B))
<=> r3_arytm_3(A,C) ) ) ) ) ).
fof(t84_arytm_3,axiom,
$true ).
fof(t85_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> r3_arytm_3(A,k10_arytm_3(A,B)) ) ) ).
fof(t86_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ~ ( k11_arytm_3(A,B) = k12_arytm_3
& A != k12_arytm_3
& B != k12_arytm_3 ) ) ) ).
fof(t87_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( r3_arytm_3(A,k11_arytm_3(B,C))
& ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ~ ( A = k11_arytm_3(B,D)
& r3_arytm_3(D,C) ) ) ) ) ) ) ).
fof(t88_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( r3_arytm_3(k11_arytm_3(B,A),k11_arytm_3(C,A))
=> ( A = k12_arytm_3
| r3_arytm_3(B,C) ) ) ) ) ) ).
fof(t89_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ~ ( k10_arytm_3(A,B) = k10_arytm_3(C,D)
& ~ r3_arytm_3(A,C)
& ~ r3_arytm_3(B,D) ) ) ) ) ) ).
fof(t90_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( r3_arytm_3(A,B)
=> r3_arytm_3(k11_arytm_3(A,C),k11_arytm_3(B,C)) ) ) ) ) ).
fof(t91_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ~ ( k11_arytm_3(A,B) = k11_arytm_3(C,D)
& ~ r3_arytm_3(A,C)
& ~ r3_arytm_3(B,D) ) ) ) ) ) ).
fof(t92_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ( A = k12_arytm_3
<=> k10_arytm_3(A,B) = B ) ) ) ).
fof(t93_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ( ( k10_arytm_3(A,B) = k10_arytm_3(C,D)
& r3_arytm_3(A,C) )
=> r3_arytm_3(D,B) ) ) ) ) ) ).
fof(t94_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( r3_arytm_3(A,B)
& r3_arytm_3(B,k10_arytm_3(A,C))
& ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ~ ( B = k10_arytm_3(A,D)
& r3_arytm_3(D,C) ) ) ) ) ) ) ).
fof(t95_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( r3_arytm_3(A,k10_arytm_3(B,C))
& ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ! [E] :
( m1_subset_1(E,k6_arytm_3)
=> ~ ( A = k10_arytm_3(D,E)
& r3_arytm_3(D,B)
& r3_arytm_3(E,C) ) ) ) ) ) ) ) ).
fof(t96_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( ~ r3_arytm_3(B,A)
& ~ r3_arytm_3(C,A)
& ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ~ ( r3_arytm_3(D,B)
& r3_arytm_3(D,C)
& ~ r3_arytm_3(D,A) ) ) ) ) ) ) ).
fof(t97_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( r3_arytm_3(A,B)
& r3_arytm_3(B,C)
& B != C
& A = C ) ) ) ) ).
fof(t98_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( ~ r3_arytm_3(k10_arytm_3(B,C),A)
& C != k12_arytm_3
& ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ! [E] :
( m1_subset_1(E,k6_arytm_3)
=> ~ ( A = k10_arytm_3(D,E)
& r3_arytm_3(D,B)
& r3_arytm_3(E,C)
& E != C ) ) ) ) ) ) ) ).
fof(t99_arytm_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k6_arytm_3)) )
=> ~ ( r2_hidden(A,k6_arytm_3)
& ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ~ ( r2_hidden(B,A)
& ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( r2_hidden(C,A)
=> r3_arytm_3(C,B) ) ) ) ) ) ) ).
fof(t100_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ~ ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( k10_arytm_3(A,C) != B
& k10_arytm_3(B,C) != A ) ) ) ) ).
fof(t101_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ~ ( ~ r3_arytm_3(B,A)
& ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( ~ r3_arytm_3(C,A)
& ~ r3_arytm_3(B,C) ) ) ) ) ) ).
fof(t102_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ~ ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> r3_arytm_3(B,A) ) ) ).
fof(t103_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ! [B] :
( m1_subset_1(B,k6_arytm_3)
=> ~ ( A != k12_arytm_3
& ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ( r2_hidden(C,k5_ordinal2)
& r3_arytm_3(B,k11_arytm_3(C,A)) ) ) ) ) ) ).
fof(s1_arytm_3,axiom,
( ( p1_s1_arytm_3(k1_xboole_0)
& ! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> ( p1_s1_arytm_3(A)
=> p1_s1_arytm_3(k1_ordinal1(A)) ) ) )
=> ! [A] :
( ( v3_ordinal1(A)
& v4_ordinal2(A) )
=> p1_s1_arytm_3(A) ) ) ).
fof(s2_arytm_3,axiom,
( ( f2_s2_arytm_3 = k13_arytm_3
& f1_s2_arytm_3 = k12_arytm_3
& r2_hidden(f3_s2_arytm_3,k5_ordinal2)
& p1_s2_arytm_3(f1_s2_arytm_3)
& ~ p1_s2_arytm_3(f3_s2_arytm_3) )
=> ? [A] :
( m1_subset_1(A,k6_arytm_3)
& r2_hidden(A,k5_ordinal2)
& p1_s2_arytm_3(A)
& ~ p1_s2_arytm_3(k10_arytm_3(A,f2_s2_arytm_3)) ) ) ).
fof(symmetry_r1_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> ( r1_arytm_3(A,B)
=> r1_arytm_3(B,A) ) ) ).
fof(reflexivity_r2_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B) )
=> r2_arytm_3(A,A) ) ).
fof(connectedness_r3_arytm_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_arytm_3)
& m1_subset_1(B,k6_arytm_3) )
=> ( r3_arytm_3(A,B)
| r3_arytm_3(B,A) ) ) ).
fof(dt_k1_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> v3_ordinal1(k1_arytm_3(A,B)) ) ).
fof(commutativity_k1_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> k1_arytm_3(A,B) = k1_arytm_3(B,A) ) ).
fof(redefinition_k1_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> k1_arytm_3(A,B) = k14_ordinal2(A,B) ) ).
fof(dt_k2_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> v3_ordinal1(k2_arytm_3(A,B)) ) ).
fof(commutativity_k2_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> k2_arytm_3(A,B) = k2_arytm_3(B,A) ) ).
fof(redefinition_k2_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> k2_arytm_3(A,B) = k15_ordinal2(A,B) ) ).
fof(dt_k3_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> m1_subset_1(k3_arytm_3(A,B),k5_ordinal2) ) ).
fof(commutativity_k3_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> k3_arytm_3(A,B) = k3_arytm_3(B,A) ) ).
fof(dt_k4_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> m1_subset_1(k4_arytm_3(A,B),k5_ordinal2) ) ).
fof(commutativity_k4_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> k4_arytm_3(A,B) = k4_arytm_3(B,A) ) ).
fof(dt_k5_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> m1_subset_1(k5_arytm_3(A,B),k5_ordinal2) ) ).
fof(dt_k6_arytm_3,axiom,
$true ).
fof(dt_k7_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> m1_subset_1(k7_arytm_3(A),k5_ordinal2) ) ).
fof(dt_k8_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> m1_subset_1(k8_arytm_3(A),k5_ordinal2) ) ).
fof(dt_k9_arytm_3,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal2(A)
& v3_ordinal1(B)
& v4_ordinal2(B) )
=> m1_subset_1(k9_arytm_3(A,B),k6_arytm_3) ) ).
fof(dt_k10_arytm_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_arytm_3)
& m1_subset_1(B,k6_arytm_3) )
=> m1_subset_1(k10_arytm_3(A,B),k6_arytm_3) ) ).
fof(commutativity_k10_arytm_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_arytm_3)
& m1_subset_1(B,k6_arytm_3) )
=> k10_arytm_3(A,B) = k10_arytm_3(B,A) ) ).
fof(dt_k11_arytm_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_arytm_3)
& m1_subset_1(B,k6_arytm_3) )
=> m1_subset_1(k11_arytm_3(A,B),k6_arytm_3) ) ).
fof(commutativity_k11_arytm_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k6_arytm_3)
& m1_subset_1(B,k6_arytm_3) )
=> k11_arytm_3(A,B) = k11_arytm_3(B,A) ) ).
fof(dt_k12_arytm_3,axiom,
( v1_xboole_0(k12_arytm_3)
& m1_subset_1(k12_arytm_3,k6_arytm_3) ) ).
fof(redefinition_k12_arytm_3,axiom,
k12_arytm_3 = k1_xboole_0 ).
fof(dt_k13_arytm_3,axiom,
( ~ v1_xboole_0(k13_arytm_3)
& v3_ordinal1(k13_arytm_3)
& m1_subset_1(k13_arytm_3,k6_arytm_3) ) ).
fof(redefinition_k13_arytm_3,axiom,
k13_arytm_3 = k4_ordinal2 ).
fof(d6_arytm_3,axiom,
k6_arytm_3 = k2_xboole_0(k4_xboole_0(a_0_0_arytm_3,a_0_1_arytm_3),k5_ordinal2) ).
fof(t81_arytm_3,axiom,
! [A] :
( m1_subset_1(A,k6_arytm_3)
=> ( r2_hidden(a_1_0_arytm_3(A),k6_arytm_3)
<=> A = k12_arytm_3 ) ) ).
fof(fraenkel_a_0_0_arytm_3,axiom,
! [A] :
( r2_hidden(A,a_0_0_arytm_3)
<=> ? [B,C] :
( m1_subset_1(B,k5_ordinal2)
& m1_subset_1(C,k5_ordinal2)
& A = k4_tarski(B,C)
& r1_arytm_3(B,C)
& C != k1_xboole_0 ) ) ).
fof(fraenkel_a_0_1_arytm_3,axiom,
! [A] :
( r2_hidden(A,a_0_1_arytm_3)
<=> ? [B] :
( m1_subset_1(B,k5_ordinal2)
& A = k4_tarski(B,k4_ordinal2) ) ) ).
fof(fraenkel_a_1_0_arytm_3,axiom,
! [A,B] :
( m1_subset_1(B,k6_arytm_3)
=> ( r2_hidden(A,a_1_0_arytm_3(B))
<=> ? [C] :
( m1_subset_1(C,k6_arytm_3)
& A = C
& ~ r3_arytm_3(B,C) ) ) ) ).
%------------------------------------------------------------------------------