SET007 Axioms: SET007+7.ax
%------------------------------------------------------------------------------
% File : SET007+7 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Subsets
% 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 : subset_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 110 ( 30 unt; 0 def)
% Number of atoms : 368 ( 48 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 316 ( 58 ~; 1 |; 47 &)
% ( 25 <=>; 185 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 14 ( 12 usr; 1 prp; 0-3 aty)
% Number of functors : 31 ( 31 usr; 7 con; 0-8 aty)
% Number of variables : 265 ( 259 !; 6 ?)
% 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_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc2_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_tarski(A)) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(fc5_subset_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C) )
=> ~ v1_xboole_0(k3_zfmisc_1(A,B,C)) ) ).
fof(fc6_subset_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D) )
=> ~ v1_xboole_0(k4_zfmisc_1(A,B,C,D)) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(d1_subset_1,axiom,
$true ).
fof(d2_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
=> ( m1_subset_1(B,A)
<=> r2_hidden(B,A) ) )
& ( v1_xboole_0(A)
=> ( m1_subset_1(B,A)
<=> v1_xboole_0(B) ) ) ) ).
fof(d3_subset_1,axiom,
! [A] : k1_subset_1(A) = k1_xboole_0 ).
fof(d4_subset_1,axiom,
! [A] : k2_subset_1(A) = A ).
fof(t1_subset_1,axiom,
$true ).
fof(t2_subset_1,axiom,
$true ).
fof(t3_subset_1,axiom,
$true ).
fof(t4_subset_1,axiom,
! [A] : m1_subset_1(k1_xboole_0,k1_zfmisc_1(A)) ).
fof(t5_subset_1,axiom,
$true ).
fof(t6_subset_1,axiom,
$true ).
fof(t7_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,B)
=> r2_hidden(D,C) ) )
=> r1_tarski(B,C) ) ) ) ).
fof(t8_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,B)
<=> r2_hidden(D,C) ) )
=> B = C ) ) ) ).
fof(t9_subset_1,axiom,
$true ).
fof(t10_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ~ ( B != k1_xboole_0
& ! [C] :
( m1_subset_1(C,A)
=> ~ r2_hidden(C,B) ) ) ) ).
fof(d5_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k3_subset_1(A,B) = k4_xboole_0(A,B) ) ).
fof(t11_subset_1,axiom,
$true ).
fof(t12_subset_1,axiom,
$true ).
fof(t13_subset_1,axiom,
$true ).
fof(t14_subset_1,axiom,
$true ).
fof(t15_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,B)
<=> ( r2_hidden(E,C)
| r2_hidden(E,D) ) ) )
=> B = k4_subset_1(A,C,D) ) ) ) ) ).
fof(t16_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,B)
<=> ( r2_hidden(E,C)
& r2_hidden(E,D) ) ) )
=> B = k5_subset_1(A,C,D) ) ) ) ) ).
fof(t17_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,B)
<=> ( r2_hidden(E,C)
& ~ r2_hidden(E,D) ) ) )
=> B = k6_subset_1(A,C,D) ) ) ) ) ).
fof(t18_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,B)
<=> ~ ( r2_hidden(E,C)
<=> r2_hidden(E,D) ) ) )
=> B = k7_subset_1(A,C,D) ) ) ) ) ).
fof(t19_subset_1,axiom,
$true ).
fof(t20_subset_1,axiom,
$true ).
fof(t21_subset_1,axiom,
! [A] : k1_subset_1(A) = k3_subset_1(A,k2_subset_1(A)) ).
fof(t22_subset_1,axiom,
! [A] : k2_subset_1(A) = k3_subset_1(A,k1_subset_1(A)) ).
fof(t23_subset_1,axiom,
$true ).
fof(t24_subset_1,axiom,
$true ).
fof(t25_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k4_subset_1(A,B,k3_subset_1(A,B)) = k2_subset_1(A) ) ).
fof(t26_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> r1_xboole_0(B,k3_subset_1(A,B)) ) ).
fof(t27_subset_1,axiom,
$true ).
fof(t28_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k4_subset_1(A,B,k2_subset_1(A)) = k2_subset_1(A) ) ).
fof(t29_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k3_subset_1(A,k4_subset_1(A,B,C)) = k5_subset_1(A,k3_subset_1(A,B),k3_subset_1(A,C)) ) ) ).
fof(t30_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k3_subset_1(A,k5_subset_1(A,B,C)) = k4_subset_1(A,k3_subset_1(A,B),k3_subset_1(A,C)) ) ) ).
fof(t31_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(B,C)
<=> r1_tarski(k3_subset_1(A,C),k3_subset_1(A,B)) ) ) ) ).
fof(t32_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k6_subset_1(A,B,C) = k5_subset_1(A,B,k3_subset_1(A,C)) ) ) ).
fof(t33_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k3_subset_1(A,k6_subset_1(A,B,C)) = k4_subset_1(A,k3_subset_1(A,B),C) ) ) ).
fof(t34_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k3_subset_1(A,k7_subset_1(A,B,C)) = k4_subset_1(A,k5_subset_1(A,B,C),k5_subset_1(A,k3_subset_1(A,B),k3_subset_1(A,C))) ) ) ).
fof(t35_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(B,k3_subset_1(A,C))
=> r1_tarski(C,k3_subset_1(A,B)) ) ) ) ).
fof(t36_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(k3_subset_1(A,B),C)
=> r1_tarski(k3_subset_1(A,C),B) ) ) ) ).
fof(t37_subset_1,axiom,
$true ).
fof(t38_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( r1_tarski(B,k3_subset_1(A,B))
<=> B = k1_subset_1(A) ) ) ).
fof(t39_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( r1_tarski(k3_subset_1(A,B),B)
<=> B = k2_subset_1(A) ) ) ).
fof(t40_subset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ( r1_tarski(B,C)
& r1_tarski(B,k3_subset_1(A,C)) )
=> B = k1_xboole_0 ) ) ).
fof(t41_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> r1_tarski(k3_subset_1(A,k4_subset_1(A,B,C)),k3_subset_1(A,B)) ) ) ).
fof(t42_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> r1_tarski(k3_subset_1(A,B),k3_subset_1(A,k5_subset_1(A,B,C))) ) ) ).
fof(t43_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,C)
<=> r1_tarski(B,k3_subset_1(A,C)) ) ) ) ).
fof(t44_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_xboole_0(B,k3_subset_1(A,C))
<=> r1_tarski(B,C) ) ) ) ).
fof(t45_subset_1,axiom,
$true ).
fof(t46_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ( r1_xboole_0(B,C)
& r1_xboole_0(k3_subset_1(A,B),k3_subset_1(A,C)) )
=> B = k3_subset_1(A,C) ) ) ) ).
fof(t47_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( r1_tarski(B,C)
& r1_xboole_0(D,C) )
=> r1_tarski(B,k3_subset_1(A,D)) ) ) ) ) ).
fof(t48_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ! [D] :
( m1_subset_1(D,B)
=> r2_hidden(D,C) )
=> r1_tarski(B,C) ) ) ) ).
fof(t49_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ! [C] :
( m1_subset_1(C,A)
=> r2_hidden(C,B) )
=> A = B ) ) ).
fof(t50_subset_1,axiom,
! [A] :
( A != k1_xboole_0
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,A)
=> ( ~ r2_hidden(C,B)
=> r2_hidden(C,k3_subset_1(A,B)) ) ) ) ) ).
fof(t51_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,B)
<=> ~ r2_hidden(D,C) ) )
=> B = k3_subset_1(A,C) ) ) ) ).
fof(t52_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ! [D] :
( m1_subset_1(D,A)
=> ( ~ r2_hidden(D,B)
<=> r2_hidden(D,C) ) )
=> B = k3_subset_1(A,C) ) ) ) ).
fof(t53_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ! [D] :
( m1_subset_1(D,A)
=> ~ ( r2_hidden(D,B)
<=> r2_hidden(D,C) ) )
=> B = k3_subset_1(A,C) ) ) ) ).
fof(t54_subset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( r2_hidden(B,k3_subset_1(A,C))
& r2_hidden(B,C) ) ) ).
fof(t55_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ( A != k1_xboole_0
=> m1_subset_1(k1_tarski(B),k1_zfmisc_1(A)) ) ) ).
fof(t56_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( A != k1_xboole_0
=> m1_subset_1(k2_tarski(B,C),k1_zfmisc_1(A)) ) ) ) ).
fof(t57_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( A != k1_xboole_0
=> m1_subset_1(k1_enumset1(B,C,D),k1_zfmisc_1(A)) ) ) ) ) ).
fof(t58_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( A != k1_xboole_0
=> m1_subset_1(k2_enumset1(B,C,D,E),k1_zfmisc_1(A)) ) ) ) ) ) ).
fof(t59_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ( A != k1_xboole_0
=> m1_subset_1(k3_enumset1(B,C,D,E,F),k1_zfmisc_1(A)) ) ) ) ) ) ) ).
fof(t60_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ( A != k1_xboole_0
=> m1_subset_1(k4_enumset1(B,C,D,E,F,G),k1_zfmisc_1(A)) ) ) ) ) ) ) ) ).
fof(t61_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,A)
=> ( A != k1_xboole_0
=> m1_subset_1(k5_enumset1(B,C,D,E,F,G,H),k1_zfmisc_1(A)) ) ) ) ) ) ) ) ) ).
fof(t62_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,A)
=> ! [I] :
( m1_subset_1(I,A)
=> ( A != k1_xboole_0
=> m1_subset_1(k6_enumset1(B,C,D,E,F,G,H,I),k1_zfmisc_1(A)) ) ) ) ) ) ) ) ) ) ).
fof(t63_subset_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(k1_tarski(A),k1_zfmisc_1(B)) ) ).
fof(d6_subset_1,axiom,
! [A] :
( ~ $true
=> ! [B] :
( m1_subset_1(B,A)
=> B = k8_subset_1(A) ) ) ).
fof(s1_subset_1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s1_subset_1))
& ! [B] :
( r2_hidden(B,A)
<=> ( r2_hidden(B,f1_s1_subset_1)
& p1_s1_subset_1(B) ) ) ) ).
fof(s2_subset_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s2_subset_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(f1_s2_subset_1))
=> ( ( ! [C] :
( m1_subset_1(C,f1_s2_subset_1)
=> ( r2_hidden(C,A)
<=> p1_s2_subset_1(C) ) )
& ! [C] :
( m1_subset_1(C,f1_s2_subset_1)
=> ( r2_hidden(C,B)
<=> p1_s2_subset_1(C) ) ) )
=> A = B ) ) ) ).
fof(s3_subset_1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s3_subset_1))
& ! [B] :
( m1_subset_1(B,f1_s3_subset_1)
=> ( r2_hidden(B,A)
<=> p1_s3_subset_1(B) ) ) ) ).
fof(s4_subset_1,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s4_subset_1)
=> ( r2_hidden(A,f2_s4_subset_1)
<=> p1_s4_subset_1(A) ) )
& ! [A] :
( m1_subset_1(A,f1_s4_subset_1)
=> ( r2_hidden(A,f3_s4_subset_1)
<=> p1_s4_subset_1(A) ) ) )
=> f2_s4_subset_1 = f3_s4_subset_1 ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(dt_m2_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m2_subset_1(C,A,B)
=> m1_subset_1(C,A) ) ) ).
fof(existence_m2_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ? [C] : m2_subset_1(C,A,B) ) ).
fof(redefinition_m2_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m2_subset_1(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(symmetry_r1_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ( r1_subset_1(A,B)
=> r1_subset_1(B,A) ) ) ).
fof(irreflexivity_r1_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ r1_subset_1(A,A) ) ).
fof(redefinition_r1_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ( r1_subset_1(A,B)
<=> r1_xboole_0(A,B) ) ) ).
fof(symmetry_r2_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ( r2_subset_1(A,B)
=> r2_subset_1(B,A) ) ) ).
fof(irreflexivity_r2_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ r2_subset_1(A,A) ) ).
fof(redefinition_r2_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ( r2_subset_1(A,B)
<=> r1_xboole_0(A,B) ) ) ).
fof(dt_k1_subset_1,axiom,
! [A] :
( v1_xboole_0(k1_subset_1(A))
& m1_subset_1(k1_subset_1(A),k1_zfmisc_1(A)) ) ).
fof(dt_k2_subset_1,axiom,
! [A] : m1_subset_1(k2_subset_1(A),k1_zfmisc_1(A)) ).
fof(dt_k3_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> m1_subset_1(k3_subset_1(A,B),k1_zfmisc_1(A)) ) ).
fof(involutiveness_k3_subset_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k3_subset_1(A,k3_subset_1(A,B)) = B ) ).
fof(dt_k4_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(commutativity_k4_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ).
fof(idempotence_k4_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k4_subset_1(A,B,B) = B ) ).
fof(redefinition_k4_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ).
fof(dt_k5_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(commutativity_k5_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ).
fof(idempotence_k5_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k5_subset_1(A,B,B) = B ) ).
fof(redefinition_k5_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k6_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_subset_1(k6_subset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(redefinition_k6_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k6_subset_1(A,B,C) = k4_xboole_0(B,C) ) ).
fof(dt_k7_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_subset_1(k7_subset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(commutativity_k7_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k7_subset_1(A,B,C) = k7_subset_1(A,C,B) ) ).
fof(redefinition_k7_subset_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k7_subset_1(A,B,C) = k5_xboole_0(B,C) ) ).
fof(dt_k8_subset_1,axiom,
! [A] : m1_subset_1(k8_subset_1(A),A) ).
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