SET007 Axioms: SET007+8.ax
%------------------------------------------------------------------------------
% File : SET007+8 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Families of Sets
% 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 : setfam_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 94 ( 36 unt; 0 def)
% Number of atoms : 224 ( 55 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 171 ( 41 ~; 3 |; 32 &)
% ( 19 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 11 ( 9 usr; 1 prp; 0-2 aty)
% Number of functors : 20 ( 20 usr; 1 con; 0-3 aty)
% Number of variables : 173 ( 164 !; 9 ?)
% 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_setfam_1,axiom,
! [A,B] :
( ( v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> v1_xboole_0(k7_setfam_1(A,B)) ) ).
fof(rc1_setfam_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_setfam_1(A) ) ).
fof(fc2_setfam_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v1_xboole_0(k1_tarski(A))
& v1_setfam_1(k1_tarski(A)) ) ) ).
fof(fc3_setfam_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ( ~ v1_xboole_0(k2_tarski(A,B))
& v1_setfam_1(k2_tarski(A,B)) ) ) ).
fof(fc4_setfam_1,axiom,
! [A,B] :
( ( v1_setfam_1(A)
& v1_setfam_1(B) )
=> v1_setfam_1(k2_xboole_0(A,B)) ) ).
fof(cc1_setfam_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_setfam_1(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ~ v1_xboole_0(B) ) ) ).
fof(rc2_setfam_1,axiom,
? [A] : ~ v2_setfam_1(A) ).
fof(cc2_setfam_1,axiom,
! [A] :
( ~ v2_setfam_1(A)
=> ~ v1_xboole_0(A) ) ).
fof(rc3_setfam_1,axiom,
! [A] :
( ~ v2_setfam_1(A)
=> ? [B] :
( m1_subset_1(B,A)
& ~ v1_xboole_0(B) ) ) ).
fof(d1_setfam_1,axiom,
! [A,B] :
( ( A != k1_xboole_0
=> ( B = k1_setfam_1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ! [D] :
( r2_hidden(D,A)
=> r2_hidden(C,D) ) ) ) )
& ( A = k1_xboole_0
=> ( B = k1_setfam_1(A)
<=> B = k1_xboole_0 ) ) ) ).
fof(t1_setfam_1,axiom,
$true ).
fof(t2_setfam_1,axiom,
k1_setfam_1(k1_xboole_0) = k1_xboole_0 ).
fof(t3_setfam_1,axiom,
! [A] : r1_tarski(k1_setfam_1(A),k3_tarski(A)) ).
fof(t4_setfam_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> r1_tarski(k1_setfam_1(B),A) ) ).
fof(t5_setfam_1,axiom,
! [A] :
( r2_hidden(k1_xboole_0,A)
=> k1_setfam_1(A) = k1_xboole_0 ) ).
fof(t6_setfam_1,axiom,
! [A,B] :
( ! [C] :
( r2_hidden(C,A)
=> r1_tarski(B,C) )
=> ( A = k1_xboole_0
| r1_tarski(B,k1_setfam_1(A)) ) ) ).
fof(t7_setfam_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> ( A = k1_xboole_0
| r1_tarski(k1_setfam_1(B),k1_setfam_1(A)) ) ) ).
fof(t8_setfam_1,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& r1_tarski(A,C) )
=> r1_tarski(k1_setfam_1(B),C) ) ).
fof(t9_setfam_1,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& r1_xboole_0(A,C) )
=> r1_xboole_0(k1_setfam_1(B),C) ) ).
fof(t10_setfam_1,axiom,
! [A,B] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& k1_setfam_1(k2_xboole_0(A,B)) != k3_xboole_0(k1_setfam_1(A),k1_setfam_1(B)) ) ).
fof(t11_setfam_1,axiom,
! [A] : k1_setfam_1(k1_tarski(A)) = A ).
fof(t12_setfam_1,axiom,
! [A,B] : k1_setfam_1(k2_tarski(A,B)) = k3_xboole_0(A,B) ).
fof(d2_setfam_1,axiom,
! [A,B] :
( r1_setfam_1(A,B)
<=> ! [C] :
~ ( r2_hidden(C,A)
& ! [D] :
~ ( r2_hidden(D,B)
& r1_tarski(C,D) ) ) ) ).
fof(d3_setfam_1,axiom,
! [A,B] :
( r2_setfam_1(A,B)
<=> ! [C] :
~ ( r2_hidden(C,B)
& ! [D] :
~ ( r2_hidden(D,A)
& r1_tarski(D,C) ) ) ) ).
fof(t13_setfam_1,axiom,
$true ).
fof(t14_setfam_1,axiom,
$true ).
fof(t15_setfam_1,axiom,
$true ).
fof(t16_setfam_1,axiom,
$true ).
fof(t17_setfam_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_setfam_1(A,B) ) ).
fof(t18_setfam_1,axiom,
! [A,B] :
( r1_setfam_1(A,B)
=> r1_tarski(k3_tarski(A),k3_tarski(B)) ) ).
fof(t19_setfam_1,axiom,
! [A,B] :
( r2_setfam_1(B,A)
=> ( A = k1_xboole_0
| r1_tarski(k1_setfam_1(B),k1_setfam_1(A)) ) ) ).
fof(t20_setfam_1,axiom,
! [A] : r1_setfam_1(k1_xboole_0,A) ).
fof(t21_setfam_1,axiom,
! [A] :
( r1_setfam_1(A,k1_xboole_0)
=> A = k1_xboole_0 ) ).
fof(t22_setfam_1,axiom,
$true ).
fof(t23_setfam_1,axiom,
! [A,B,C] :
( ( r1_setfam_1(A,B)
& r1_setfam_1(B,C) )
=> r1_setfam_1(A,C) ) ).
fof(t24_setfam_1,axiom,
! [A,B] :
( r1_setfam_1(B,k1_tarski(A))
=> ! [C] :
( r2_hidden(C,B)
=> r1_tarski(C,A) ) ) ).
fof(t25_setfam_1,axiom,
! [A,B,C] :
( r1_setfam_1(C,k2_tarski(A,B))
=> ! [D] :
~ ( r2_hidden(D,C)
& ~ r1_tarski(D,A)
& ~ r1_tarski(D,B) ) ) ).
fof(d4_setfam_1,axiom,
! [A,B,C] :
( C = k2_setfam_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E,F] :
( r2_hidden(E,A)
& r2_hidden(F,B)
& D = k2_xboole_0(E,F) ) ) ) ).
fof(d5_setfam_1,axiom,
! [A,B,C] :
( C = k3_setfam_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E,F] :
( r2_hidden(E,A)
& r2_hidden(F,B)
& D = k3_xboole_0(E,F) ) ) ) ).
fof(d6_setfam_1,axiom,
! [A,B,C] :
( C = k4_setfam_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E,F] :
( r2_hidden(E,A)
& r2_hidden(F,B)
& D = k4_xboole_0(E,F) ) ) ) ).
fof(t26_setfam_1,axiom,
$true ).
fof(t27_setfam_1,axiom,
$true ).
fof(t28_setfam_1,axiom,
$true ).
fof(t29_setfam_1,axiom,
! [A] : r1_setfam_1(A,k2_setfam_1(A,A)) ).
fof(t30_setfam_1,axiom,
! [A] : r1_setfam_1(k3_setfam_1(A,A),A) ).
fof(t31_setfam_1,axiom,
! [A] : r1_setfam_1(k4_setfam_1(A,A),A) ).
fof(t32_setfam_1,axiom,
$true ).
fof(t33_setfam_1,axiom,
$true ).
fof(t34_setfam_1,axiom,
! [A,B] :
( ~ r1_xboole_0(A,B)
=> k3_xboole_0(k1_setfam_1(A),k1_setfam_1(B)) = k1_setfam_1(k3_setfam_1(A,B)) ) ).
fof(t35_setfam_1,axiom,
! [A,B] :
( B != k1_xboole_0
=> k2_xboole_0(A,k1_setfam_1(B)) = k1_setfam_1(k2_setfam_1(k1_tarski(A),B)) ) ).
fof(t36_setfam_1,axiom,
! [A,B] : k3_xboole_0(A,k3_tarski(B)) = k3_tarski(k3_setfam_1(k1_tarski(A),B)) ).
fof(t37_setfam_1,axiom,
! [A,B] :
( B != k1_xboole_0
=> k4_xboole_0(A,k3_tarski(B)) = k1_setfam_1(k4_setfam_1(k1_tarski(A),B)) ) ).
fof(t38_setfam_1,axiom,
! [A,B] :
( B != k1_xboole_0
=> k4_xboole_0(A,k1_setfam_1(B)) = k3_tarski(k4_setfam_1(k1_tarski(A),B)) ) ).
fof(t39_setfam_1,axiom,
! [A,B] : k3_tarski(k3_setfam_1(A,B)) = k3_xboole_0(k3_tarski(A),k3_tarski(B)) ).
fof(t40_setfam_1,axiom,
! [A,B] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& ~ r1_tarski(k2_xboole_0(k1_setfam_1(A),k1_setfam_1(B)),k1_setfam_1(k2_setfam_1(A,B))) ) ).
fof(t41_setfam_1,axiom,
! [A,B] : r1_tarski(k1_setfam_1(k4_setfam_1(A,B)),k4_xboole_0(k1_setfam_1(A),k1_setfam_1(B))) ).
fof(t42_setfam_1,axiom,
$true ).
fof(t43_setfam_1,axiom,
$true ).
fof(t44_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,B)
<=> r2_hidden(D,C) ) )
=> B = C ) ) ) ).
fof(d7_setfam_1,axiom,
$true ).
fof(d8_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( C = k7_setfam_1(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,C)
<=> r2_hidden(k3_subset_1(A,D),B) ) ) ) ) ) ).
fof(t45_setfam_1,axiom,
$true ).
fof(t46_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ~ ( B != k1_xboole_0
& k7_setfam_1(A,B) = k1_xboole_0 ) ) ).
fof(t47_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( B != k1_xboole_0
=> k6_subset_1(A,k2_subset_1(A),k5_setfam_1(A,B)) = k6_setfam_1(A,k7_setfam_1(A,B)) ) ) ).
fof(t48_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( B != k1_xboole_0
=> k5_setfam_1(A,k7_setfam_1(A,B)) = k6_subset_1(A,k2_subset_1(A),k6_setfam_1(A,B)) ) ) ).
fof(t49_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(k3_subset_1(A,C),k7_setfam_1(A,B))
<=> r2_hidden(C,B) ) ) ) ).
fof(t50_setfam_1,axiom,
$true ).
fof(t51_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r1_tarski(k7_setfam_1(A,B),k7_setfam_1(A,C))
=> r1_tarski(B,C) ) ) ) ).
fof(t52_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r1_tarski(k7_setfam_1(A,B),C)
<=> r1_tarski(B,k7_setfam_1(A,C)) ) ) ) ).
fof(t53_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( k7_setfam_1(A,B) = k7_setfam_1(A,C)
=> B = C ) ) ) ).
fof(t54_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k7_setfam_1(A,k4_subset_1(k1_zfmisc_1(A),B,C)) = k4_subset_1(k1_zfmisc_1(A),k7_setfam_1(A,B),k7_setfam_1(A,C)) ) ) ).
fof(t55_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( B = k1_tarski(A)
=> k7_setfam_1(A,B) = k1_tarski(k1_xboole_0) ) ) ).
fof(d9_setfam_1,axiom,
! [A] :
( v1_setfam_1(A)
<=> ~ r2_hidden(k1_xboole_0,A) ) ).
fof(t56_setfam_1,axiom,
! [A,B,C] :
( ( r1_tarski(k3_tarski(A),B)
& r2_hidden(C,A) )
=> r1_tarski(C,B) ) ).
fof(t57_setfam_1,axiom,
! [A,B,C] :
( ( r1_tarski(C,k3_tarski(k2_xboole_0(A,B)))
& ! [D] :
( r2_hidden(D,B)
=> r1_xboole_0(D,C) ) )
=> r1_tarski(C,k3_tarski(A)) ) ).
fof(d10_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( B != k1_xboole_0
=> k8_setfam_1(A,B) = k6_setfam_1(A,B) )
& ( B = k1_xboole_0
=> k8_setfam_1(A,B) = A ) ) ) ).
fof(t58_setfam_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r2_hidden(B,A)
=> ( r2_hidden(B,k8_setfam_1(A,C))
<=> ! [D] :
( r2_hidden(D,C)
=> r2_hidden(B,D) ) ) ) ) ).
fof(t59_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r1_tarski(B,C)
=> r1_tarski(k8_setfam_1(A,C),k8_setfam_1(A,B)) ) ) ) ).
fof(d11_setfam_1,axiom,
! [A] :
( v2_setfam_1(A)
<=> ! [B] :
( ~ v1_xboole_0(B)
=> ~ r2_hidden(B,A) ) ) ).
fof(reflexivity_r1_setfam_1,axiom,
! [A,B] : r1_setfam_1(A,A) ).
fof(reflexivity_r2_setfam_1,axiom,
! [A,B] : r2_setfam_1(B,B) ).
fof(dt_k1_setfam_1,axiom,
$true ).
fof(dt_k2_setfam_1,axiom,
$true ).
fof(commutativity_k2_setfam_1,axiom,
! [A,B] : k2_setfam_1(A,B) = k2_setfam_1(B,A) ).
fof(dt_k3_setfam_1,axiom,
$true ).
fof(commutativity_k3_setfam_1,axiom,
! [A,B] : k3_setfam_1(A,B) = k3_setfam_1(B,A) ).
fof(dt_k4_setfam_1,axiom,
$true ).
fof(dt_k5_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k5_setfam_1(A,B),k1_zfmisc_1(A)) ) ).
fof(redefinition_k5_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k5_setfam_1(A,B) = k3_tarski(B) ) ).
fof(dt_k6_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k6_setfam_1(A,B),k1_zfmisc_1(A)) ) ).
fof(redefinition_k6_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k6_setfam_1(A,B) = k1_setfam_1(B) ) ).
fof(dt_k7_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k7_setfam_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(involutiveness_k7_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k7_setfam_1(A,k7_setfam_1(A,B)) = B ) ).
fof(dt_k8_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k8_setfam_1(A,B),k1_zfmisc_1(A)) ) ).
%------------------------------------------------------------------------------