SET007 Axioms: SET007+60.ax
%------------------------------------------------------------------------------
% File : SET007+60 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Equivalence Relations and Classes of Abstraction
% 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 : eqrel_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 80 ( 16 unt; 0 def)
% Number of atoms : 453 ( 51 equ)
% Maximal formula atoms : 20 ( 5 avg)
% Number of connectives : 398 ( 25 ~; 2 |; 261 &)
% ( 11 <=>; 99 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 27 ( 27 usr; 5 con; 0-4 aty)
% Number of variables : 199 ( 191 !; 8 ?)
% 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_eqrel_1,axiom,
! [A] :
( v1_relat_1(k1_eqrel_1(A))
& v1_relat_2(k1_eqrel_1(A))
& v1_partfun1(k1_eqrel_1(A),A,A) ) ).
fof(fc2_eqrel_1,axiom,
! [A] :
( v1_relat_1(k1_eqrel_1(A))
& v1_relat_2(k1_eqrel_1(A))
& v3_relat_2(k1_eqrel_1(A))
& v8_relat_2(k1_eqrel_1(A))
& v1_partfun1(k1_eqrel_1(A),A,A) ) ).
fof(cc1_eqrel_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ~ v1_xboole_0(B) ) ) ).
fof(cc2_eqrel_1,axiom,
! [A,B] :
( m1_eqrel_1(B,A)
=> v1_setfam_1(B) ) ).
fof(fc3_eqrel_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> ~ v1_xboole_0(k7_eqrel_1(A,B)) ) ).
fof(fc4_eqrel_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> ( ~ v1_xboole_0(k7_eqrel_1(A,B))
& v1_setfam_1(k7_eqrel_1(A,B)) ) ) ).
fof(d1_eqrel_1,axiom,
! [A] : k1_eqrel_1(A) = k2_zfmisc_1(A,A) ).
fof(t1_eqrel_1,axiom,
$true ).
fof(t2_eqrel_1,axiom,
$true ).
fof(t3_eqrel_1,axiom,
$true ).
fof(t4_eqrel_1,axiom,
! [A] :
( r1_relat_2(k6_partfun1(A),A)
& r3_relat_2(k6_partfun1(A),A)
& r8_relat_2(k6_partfun1(A),A) ) ).
fof(t5_eqrel_1,axiom,
$true ).
fof(t6_eqrel_1,axiom,
! [A] :
( v3_relat_2(k6_partfun1(A))
& v8_relat_2(k6_partfun1(A))
& v1_partfun1(k6_partfun1(A),A,A)
& m2_relset_1(k6_partfun1(A),A,A) ) ).
fof(t7_eqrel_1,axiom,
! [A] :
( v3_relat_2(k1_eqrel_1(A))
& v8_relat_2(k1_eqrel_1(A))
& v1_partfun1(k1_eqrel_1(A),A,A)
& m2_relset_1(k1_eqrel_1(A),A,A) ) ).
fof(t8_eqrel_1,axiom,
$true ).
fof(t9_eqrel_1,axiom,
$true ).
fof(t10_eqrel_1,axiom,
$true ).
fof(t11_eqrel_1,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r2_hidden(B,A)
=> r2_hidden(k4_tarski(B,B),C) ) ) ).
fof(t12_eqrel_1,axiom,
! [A,B,C,D] :
( ( v3_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ( r2_hidden(k4_tarski(B,C),D)
=> r2_hidden(k4_tarski(C,B),D) ) ) ).
fof(t13_eqrel_1,axiom,
! [A,B,C,D,E] :
( ( v8_relat_2(E)
& v1_partfun1(E,A,A)
& m2_relset_1(E,A,A) )
=> ( ( r2_hidden(k4_tarski(B,C),E)
& r2_hidden(k4_tarski(C,D),E) )
=> r2_hidden(k4_tarski(B,D),E) ) ) ).
fof(t14_eqrel_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ~ ( ? [C] : r2_hidden(C,A)
& B = k1_xboole_0 ) ) ).
fof(t15_eqrel_1,axiom,
$true ).
fof(t16_eqrel_1,axiom,
! [A,B] :
( m2_relset_1(B,A,A)
=> ( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
<=> ( v1_relat_2(B)
& v3_relat_2(B)
& v8_relat_2(B)
& k3_relat_1(B) = A ) ) ) ).
fof(t17_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k4_eqrel_1(A,k6_partfun1(A),B) = k6_partfun1(A) ) ).
fof(t18_eqrel_1,axiom,
! [A,B] :
( m2_relset_1(B,A,A)
=> k2_eqrel_1(A,k1_eqrel_1(A),B) = B ) ).
fof(t19_eqrel_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(A,A))))
=> ( ! [C] :
( r2_hidden(C,B)
=> ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) ) )
=> ( B = k1_xboole_0
| ( v3_relat_2(k6_setfam_1(k2_zfmisc_1(A,A),B))
& v8_relat_2(k6_setfam_1(k2_zfmisc_1(A,A),B))
& v1_partfun1(k6_setfam_1(k2_zfmisc_1(A,A),B),A,A)
& m2_relset_1(k6_setfam_1(k2_zfmisc_1(A,A),B),A,A) ) ) ) ) ).
fof(t20_eqrel_1,axiom,
! [A,B] :
( m2_relset_1(B,A,A)
=> ? [C] :
( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A)
& r1_tarski(B,C)
& ! [D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ( r1_tarski(B,D)
=> r1_tarski(C,D) ) ) ) ) ).
fof(d2_eqrel_1,axiom,
$true ).
fof(d3_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ( D = k5_eqrel_1(A,B,C)
<=> ( r1_tarski(k3_eqrel_1(A,B,C),D)
& ! [E] :
( ( v3_relat_2(E)
& v8_relat_2(E)
& v1_partfun1(E,A,A)
& m2_relset_1(E,A,A) )
=> ( r1_tarski(k3_eqrel_1(A,B,C),E)
=> r1_tarski(D,E) ) ) ) ) ) ) ) ).
fof(t21_eqrel_1,axiom,
$true ).
fof(t22_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k5_eqrel_1(A,B,B) = B ) ).
fof(t23_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> k5_eqrel_1(A,B,C) = k5_eqrel_1(A,C,B) ) ) ).
fof(t24_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> k4_eqrel_1(A,B,k5_eqrel_1(A,B,C)) = B ) ) ).
fof(t25_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> k5_eqrel_1(A,B,k4_eqrel_1(A,B,C)) = B ) ) ).
fof(d4_eqrel_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] : k6_eqrel_1(A,B,C) = k10_relset_1(A,A,B,k1_tarski(C)) ) ).
fof(t26_eqrel_1,axiom,
$true ).
fof(t27_eqrel_1,axiom,
! [A,B,C,D] :
( ( v1_relat_2(D)
& v3_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ( r2_hidden(B,k6_eqrel_1(A,D,C))
<=> r2_hidden(k4_tarski(B,C),D) ) ) ).
fof(t28_eqrel_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( r2_hidden(C,A)
=> r2_hidden(C,k6_eqrel_1(A,B,C)) ) ) ).
fof(t29_eqrel_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
~ ( r2_hidden(C,A)
& ! [D] : ~ r2_hidden(C,k6_eqrel_1(A,B,D)) ) ) ).
fof(t30_eqrel_1,axiom,
! [A,B,C,D,E] :
( ( v1_relat_2(E)
& v3_relat_2(E)
& v8_relat_2(E)
& v1_partfun1(E,A,A)
& m2_relset_1(E,A,A) )
=> ( ( r2_hidden(B,k6_eqrel_1(A,E,C))
& r2_hidden(D,k6_eqrel_1(A,E,C)) )
=> r2_hidden(k4_tarski(B,D),E) ) ) ).
fof(t31_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( r2_hidden(D,A)
=> ( r2_hidden(B,k6_eqrel_1(A,C,D))
<=> k6_eqrel_1(A,C,D) = k6_eqrel_1(A,C,B) ) ) ) ).
fof(t32_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C,D] :
~ ( r2_hidden(C,A)
& r2_hidden(D,A)
& k6_eqrel_1(A,B,C) != k6_eqrel_1(A,B,D)
& ~ r1_xboole_0(k6_eqrel_1(A,B,C),k6_eqrel_1(A,B,D)) ) ) ).
fof(t33_eqrel_1,axiom,
! [A,B] :
( r2_hidden(B,A)
=> k6_eqrel_1(A,k6_partfun1(A),B) = k1_tarski(B) ) ).
fof(t34_eqrel_1,axiom,
! [A,B] :
( r2_hidden(B,A)
=> k6_eqrel_1(A,k1_eqrel_1(A),B) = A ) ).
fof(t35_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ( ? [C] : k6_eqrel_1(A,B,C) = A
=> B = k1_eqrel_1(A) ) ) ).
fof(t36_eqrel_1,axiom,
! [A,B,C,D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,B,B)
& m2_relset_1(D,B,B) )
=> ! [E] :
( ( v3_relat_2(E)
& v8_relat_2(E)
& v1_partfun1(E,B,B)
& m2_relset_1(E,B,B) )
=> ( r2_hidden(A,B)
=> ( r2_hidden(k4_tarski(A,C),k5_eqrel_1(B,D,E))
<=> ? [F] :
( v1_relat_1(F)
& v1_funct_1(F)
& v1_finseq_1(F)
& r1_xreal_0(np__1,k3_finseq_1(F))
& A = k1_funct_1(F,np__1)
& C = k1_funct_1(F,k3_finseq_1(F))
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,G)
=> ( r1_xreal_0(k3_finseq_1(F),G)
| r2_hidden(k4_tarski(k1_funct_1(F,G),k1_funct_1(F,k1_nat_1(G,np__1))),k3_eqrel_1(B,D,E)) ) ) ) ) ) ) ) ) ).
fof(t37_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ( D = k3_eqrel_1(A,B,C)
=> ! [E] :
~ ( r2_hidden(E,A)
& k6_eqrel_1(A,D,E) != k6_eqrel_1(A,B,E)
& k6_eqrel_1(A,D,E) != k6_eqrel_1(A,C,E) ) ) ) ) ) ).
fof(t38_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ~ ( k3_eqrel_1(A,B,C) = k1_eqrel_1(A)
& B != k1_eqrel_1(A)
& C != k1_eqrel_1(A) ) ) ) ).
fof(d5_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( C = k7_eqrel_1(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,C)
<=> ? [E] :
( r2_hidden(E,A)
& D = k6_eqrel_1(A,B,E) ) ) ) ) ) ) ).
fof(t39_eqrel_1,axiom,
$true ).
fof(t40_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ( A = k1_xboole_0
=> k7_eqrel_1(A,B) = k1_xboole_0 ) ) ).
fof(d6_eqrel_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( A != k1_xboole_0
=> ( m1_eqrel_1(B,A)
<=> ( k5_setfam_1(A,B) = A
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
=> ( C != k1_xboole_0
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( r2_hidden(D,B)
& C != D
& ~ r1_xboole_0(C,D) ) ) ) ) ) ) ) )
& ( A = k1_xboole_0
=> ( m1_eqrel_1(B,A)
<=> B = k1_xboole_0 ) ) ) ) ).
fof(t41_eqrel_1,axiom,
$true ).
fof(t42_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> m1_eqrel_1(k7_eqrel_1(A,B),A) ) ).
fof(t43_eqrel_1,axiom,
! [A,B] :
( m1_eqrel_1(B,A)
=> ? [C] :
( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A)
& B = k7_eqrel_1(A,C) ) ) ).
fof(t44_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( r2_hidden(D,A)
=> ( r2_hidden(k4_tarski(D,B),C)
<=> k6_eqrel_1(A,C,D) = k6_eqrel_1(A,C,B) ) ) ) ).
fof(t45_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,B,B)
& m2_relset_1(C,B,B) )
=> ~ ( r2_hidden(A,k7_eqrel_1(B,C))
& ! [D] :
( m1_subset_1(D,B)
=> A != k6_eqrel_1(B,C,D) ) ) ) ).
fof(s1_eqrel_1,axiom,
( ( ! [A] :
( r2_hidden(A,f1_s1_eqrel_1)
=> p1_s1_eqrel_1(A,A) )
& ! [A,B] :
( p1_s1_eqrel_1(A,B)
=> p1_s1_eqrel_1(B,A) )
& ! [A,B,C] :
( ( p1_s1_eqrel_1(A,B)
& p1_s1_eqrel_1(B,C) )
=> p1_s1_eqrel_1(A,C) ) )
=> ? [A] :
( v3_relat_2(A)
& v8_relat_2(A)
& v1_partfun1(A,f1_s1_eqrel_1,f1_s1_eqrel_1)
& m2_relset_1(A,f1_s1_eqrel_1,f1_s1_eqrel_1)
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),A)
<=> ( r2_hidden(B,f1_s1_eqrel_1)
& r2_hidden(C,f1_s1_eqrel_1)
& p1_s1_eqrel_1(B,C) ) ) ) ) ).
fof(dt_m1_eqrel_1,axiom,
! [A,B] :
( m1_eqrel_1(B,A)
=> m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(existence_m1_eqrel_1,axiom,
! [A] :
? [B] : m1_eqrel_1(B,A) ).
fof(dt_k1_eqrel_1,axiom,
! [A] : m2_relset_1(k1_eqrel_1(A),A,A) ).
fof(dt_k2_eqrel_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& m1_relset_1(C,A,A) )
=> m2_relset_1(k2_eqrel_1(A,B,C),A,A) ) ).
fof(commutativity_k2_eqrel_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& m1_relset_1(C,A,A) )
=> k2_eqrel_1(A,B,C) = k2_eqrel_1(A,C,B) ) ).
fof(idempotence_k2_eqrel_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& m1_relset_1(C,A,A) )
=> k2_eqrel_1(A,B,B) = B ) ).
fof(redefinition_k2_eqrel_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& m1_relset_1(C,A,A) )
=> k2_eqrel_1(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k3_eqrel_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& m1_relset_1(C,A,A) )
=> m2_relset_1(k3_eqrel_1(A,B,C),A,A) ) ).
fof(commutativity_k3_eqrel_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& m1_relset_1(C,A,A) )
=> k3_eqrel_1(A,B,C) = k3_eqrel_1(A,C,B) ) ).
fof(idempotence_k3_eqrel_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& m1_relset_1(C,A,A) )
=> k3_eqrel_1(A,B,B) = B ) ).
fof(redefinition_k3_eqrel_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& m1_relset_1(C,A,A) )
=> k3_eqrel_1(A,B,C) = k2_xboole_0(B,C) ) ).
fof(dt_k4_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m1_relset_1(C,A,A) )
=> ( v3_relat_2(k4_eqrel_1(A,B,C))
& v8_relat_2(k4_eqrel_1(A,B,C))
& v1_partfun1(k4_eqrel_1(A,B,C),A,A)
& m2_relset_1(k4_eqrel_1(A,B,C),A,A) ) ) ).
fof(commutativity_k4_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m1_relset_1(C,A,A) )
=> k4_eqrel_1(A,B,C) = k4_eqrel_1(A,C,B) ) ).
fof(idempotence_k4_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m1_relset_1(C,A,A) )
=> k4_eqrel_1(A,B,B) = B ) ).
fof(redefinition_k4_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m1_relset_1(C,A,A) )
=> k4_eqrel_1(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k5_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m1_relset_1(C,A,A) )
=> ( v3_relat_2(k5_eqrel_1(A,B,C))
& v8_relat_2(k5_eqrel_1(A,B,C))
& v1_partfun1(k5_eqrel_1(A,B,C),A,A)
& m2_relset_1(k5_eqrel_1(A,B,C),A,A) ) ) ).
fof(commutativity_k5_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m1_relset_1(C,A,A) )
=> k5_eqrel_1(A,B,C) = k5_eqrel_1(A,C,B) ) ).
fof(idempotence_k5_eqrel_1,axiom,
! [A,B,C] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m1_relset_1(C,A,A) )
=> k5_eqrel_1(A,B,B) = B ) ).
fof(dt_k6_eqrel_1,axiom,
! [A,B,C] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> m1_subset_1(k6_eqrel_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(dt_k7_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> m1_subset_1(k7_eqrel_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(dt_k8_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> m1_eqrel_1(k8_eqrel_1(A,B),A) ) ).
fof(redefinition_k8_eqrel_1,axiom,
! [A,B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> k8_eqrel_1(A,B) = k7_eqrel_1(A,B) ) ).
%------------------------------------------------------------------------------