SET007 Axioms: SET007+16.ax
%------------------------------------------------------------------------------
% File : SET007+16 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Relations Defined on 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 : relset_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 89 ( 20 unt; 0 def)
% Number of atoms : 253 ( 34 equ)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 193 ( 29 ~; 0 |; 42 &)
% ( 11 <=>; 111 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 11 ( 9 usr; 1 prp; 0-3 aty)
% Number of functors : 33 ( 33 usr; 5 con; 0-6 aty)
% Number of variables : 275 ( 266 !; 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(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(d1_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
<=> r1_tarski(C,k2_zfmisc_1(A,B)) ) ).
fof(t1_relset_1,axiom,
$true ).
fof(t2_relset_1,axiom,
$true ).
fof(t3_relset_1,axiom,
$true ).
fof(t4_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,B,C)
=> ( r1_tarski(A,D)
=> m2_relset_1(A,B,C) ) ) ).
fof(t5_relset_1,axiom,
$true ).
fof(t6_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,B)
=> ~ ( r2_hidden(C,D)
& ! [E,F] :
~ ( C = k4_tarski(E,F)
& r2_hidden(E,A)
& r2_hidden(F,B) ) ) ) ).
fof(t7_relset_1,axiom,
$true ).
fof(t8_relset_1,axiom,
! [A,B,C,D] :
( ( r2_hidden(C,A)
& r2_hidden(D,B) )
=> m2_relset_1(k1_tarski(k4_tarski(C,D)),A,B) ) ).
fof(t9_relset_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r1_tarski(k1_relat_1(B),A)
=> m2_relset_1(B,A,k2_relat_1(B)) ) ) ).
fof(t10_relset_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r1_tarski(k2_relat_1(B),A)
=> m2_relset_1(B,k1_relat_1(B),A) ) ) ).
fof(t11_relset_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( r1_tarski(k1_relat_1(C),A)
& r1_tarski(k2_relat_1(C),B) )
=> m2_relset_1(C,A,B) ) ) ).
fof(t12_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( r1_tarski(k1_relat_1(C),A)
& r1_tarski(k2_relat_1(C),B) ) ) ).
fof(t13_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,C)
=> ( r1_tarski(k1_relat_1(D),B)
=> m2_relset_1(D,B,C) ) ) ).
fof(t14_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,C,A)
=> ( r1_tarski(k2_relat_1(D),B)
=> m2_relset_1(D,C,B) ) ) ).
fof(t15_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,C)
=> ( r1_tarski(A,B)
=> m2_relset_1(D,B,C) ) ) ).
fof(t16_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,C,A)
=> ( r1_tarski(A,B)
=> m2_relset_1(D,C,B) ) ) ).
fof(t17_relset_1,axiom,
! [A,B,C,D,E] :
( m2_relset_1(E,A,C)
=> ( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> m2_relset_1(E,B,D) ) ) ).
fof(t18_relset_1,axiom,
$true ).
fof(t19_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> r1_tarski(k3_relat_1(C),k2_xboole_0(A,B)) ) ).
fof(t20_relset_1,axiom,
$true ).
fof(t21_relset_1,axiom,
$true ).
fof(t22_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,B,A)
=> ( ! [D] :
~ ( r2_hidden(D,B)
& ! [E] : ~ r2_hidden(k4_tarski(D,E),C) )
<=> k4_relset_1(B,A,C) = B ) ) ).
fof(t23_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( ! [D] :
~ ( r2_hidden(D,B)
& ! [E] : ~ r2_hidden(k4_tarski(E,D),C) )
<=> k5_relset_1(A,B,C) = B ) ) ).
fof(t24_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( k4_relset_1(B,A,k6_relset_1(A,B,C)) = k5_relset_1(A,B,C)
& k5_relset_1(B,A,k6_relset_1(A,B,C)) = k4_relset_1(A,B,C) ) ) ).
fof(t25_relset_1,axiom,
! [A,B] : m2_relset_1(k1_xboole_0,A,B) ).
fof(t26_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( m2_relset_1(C,k1_xboole_0,B)
=> C = k1_xboole_0 ) ) ).
fof(t27_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,B,A)
=> ( m2_relset_1(C,B,k1_xboole_0)
=> C = k1_xboole_0 ) ) ).
fof(t28_relset_1,axiom,
! [A] : r1_tarski(k6_relat_1(A),k2_zfmisc_1(A,A)) ).
fof(t29_relset_1,axiom,
! [A] : m2_relset_1(k6_relat_1(A),A,A) ).
fof(t30_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,B)
=> ( r1_tarski(k6_relat_1(C),D)
=> ( r1_tarski(C,k4_relset_1(A,B,D))
& r1_tarski(C,k5_relset_1(A,B,D)) ) ) ) ).
fof(t31_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,B,A)
=> ( r1_tarski(k6_relat_1(B),C)
=> ( B = k4_relset_1(B,A,C)
& r1_tarski(B,k5_relset_1(B,A,C)) ) ) ) ).
fof(t32_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( r1_tarski(k6_relat_1(B),C)
=> ( r1_tarski(B,k4_relset_1(A,B,C))
& B = k5_relset_1(A,B,C) ) ) ) ).
fof(t33_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,C)
=> m2_relset_1(k8_relset_1(A,C,D,B),B,C) ) ).
fof(t34_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,B,A)
=> ( r1_tarski(B,C)
=> k8_relset_1(B,A,D,C) = D ) ) ).
fof(t35_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,C,A)
=> m2_relset_1(k9_relset_1(C,A,B,D),C,B) ) ).
fof(t36_relset_1,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,B)
=> ( r1_tarski(B,C)
=> k9_relset_1(A,B,C,D) = D ) ) ).
fof(t37_relset_1,axiom,
$true ).
fof(t38_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( k10_relset_1(A,B,C,A) = k5_relset_1(A,B,C)
& k11_relset_1(A,B,C,B) = k4_relset_1(A,B,C) ) ) ).
fof(t39_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,B,A)
=> ( k10_relset_1(B,A,C,k11_relset_1(B,A,C,A)) = k5_relset_1(B,A,C)
& k11_relset_1(B,A,C,k10_relset_1(B,A,C,B)) = k4_relset_1(B,A,C) ) ) ).
fof(t40_relset_1,axiom,
$true ).
fof(t41_relset_1,axiom,
$true ).
fof(t42_relset_1,axiom,
$true ).
fof(t43_relset_1,axiom,
$true ).
fof(t44_relset_1,axiom,
$true ).
fof(t45_relset_1,axiom,
! [A,B] :
( m2_relset_1(B,A,A)
=> ( k5_relat_1(B,k6_relat_1(A)) = B
& k5_relat_1(k6_relat_1(A),B) = B ) ) ).
fof(t46_relset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k6_relat_1(A) != k1_xboole_0 ) ).
fof(t47_relset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,k4_relset_1(A,B,C))
<=> ? [E] :
( m1_subset_1(E,B)
& r2_hidden(k4_tarski(D,E),C) ) ) ) ) ) ) ).
fof(t48_relset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,B,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,k5_relset_1(B,A,C))
<=> ? [E] :
( m1_subset_1(E,B)
& r2_hidden(k4_tarski(E,D),C) ) ) ) ) ) ) ).
fof(t49_relset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,A)
=> ~ ( r2_hidden(D,k4_relset_1(A,B,C))
& ! [E] :
( m1_subset_1(E,B)
=> ~ r2_hidden(E,k5_relset_1(A,B,C)) ) ) ) ) ) ) ).
fof(t50_relset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,B,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ~ ( r2_hidden(D,k5_relset_1(B,A,C))
& ! [E] :
( m1_subset_1(E,B)
=> ~ r2_hidden(E,k4_relset_1(B,A,C)) ) ) ) ) ) ) ).
fof(t51_relset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_relset_1(D,A,B)
=> ! [E] :
( m2_relset_1(E,B,C)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,C)
=> ( r2_hidden(k4_tarski(F,G),k7_relset_1(A,B,B,C,D,E))
<=> ? [H] :
( m1_subset_1(H,B)
& r2_hidden(k4_tarski(F,H),D)
& r2_hidden(k4_tarski(H,G),E) ) ) ) ) ) ) ) ) ) ).
fof(t52_relset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_relset_1(D,C,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,k10_relset_1(C,A,D,B))
<=> ? [F] :
( m1_subset_1(F,C)
& r2_hidden(k4_tarski(F,E),D)
& r2_hidden(F,B) ) ) ) ) ) ) ) ).
fof(t53_relset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_relset_1(D,A,C)
=> ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,k11_relset_1(A,C,D,B))
<=> ? [F] :
( m1_subset_1(F,C)
& r2_hidden(k4_tarski(E,F),D)
& r2_hidden(F,B) ) ) ) ) ) ) ) ).
fof(s1_relset_1,axiom,
? [A] :
( m2_relset_1(A,f1_s1_relset_1,f2_s1_relset_1)
& ! [B,C] :
( r2_hidden(k4_tarski(B,C),A)
<=> ( r2_hidden(B,f1_s1_relset_1)
& r2_hidden(C,f2_s1_relset_1)
& p1_s1_relset_1(B,C) ) ) ) ).
fof(s2_relset_1,axiom,
? [A] :
( m2_relset_1(A,f1_s2_relset_1,f2_s2_relset_1)
& ! [B] :
( m1_subset_1(B,f1_s2_relset_1)
=> ! [C] :
( m1_subset_1(C,f2_s2_relset_1)
=> ( r2_hidden(k4_tarski(B,C),A)
<=> p1_s2_relset_1(B,C) ) ) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(dt_k1_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> m2_relset_1(k1_relset_1(A,B,C,D),A,B) ) ).
fof(commutativity_k1_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> k1_relset_1(A,B,C,D) = k1_relset_1(A,B,D,C) ) ).
fof(idempotence_k1_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> k1_relset_1(A,B,C,C) = C ) ).
fof(redefinition_k1_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> k1_relset_1(A,B,C,D) = k2_xboole_0(C,D) ) ).
fof(dt_k2_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> m2_relset_1(k2_relset_1(A,B,C,D),A,B) ) ).
fof(commutativity_k2_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> k2_relset_1(A,B,C,D) = k2_relset_1(A,B,D,C) ) ).
fof(idempotence_k2_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> k2_relset_1(A,B,C,C) = C ) ).
fof(redefinition_k2_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> k2_relset_1(A,B,C,D) = k3_xboole_0(C,D) ) ).
fof(dt_k3_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> m2_relset_1(k3_relset_1(A,B,C,D),A,B) ) ).
fof(redefinition_k3_relset_1,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_relset_1(D,A,B) )
=> k3_relset_1(A,B,C,D) = k4_xboole_0(C,D) ) ).
fof(dt_k4_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(redefinition_k4_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k4_relset_1(A,B,C) = k1_relat_1(C) ) ).
fof(dt_k5_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) ).
fof(redefinition_k5_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k5_relset_1(A,B,C) = k2_relat_1(C) ) ).
fof(dt_k6_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> m2_relset_1(k6_relset_1(A,B,C),B,A) ) ).
fof(involutiveness_k6_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k6_relset_1(A,B,k6_relset_1(A,B,C)) = C ) ).
fof(redefinition_k6_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k6_relset_1(A,B,C) = k4_relat_1(C) ) ).
fof(dt_k7_relset_1,axiom,
! [A,B,C,D,E,F] :
( ( m1_relset_1(E,A,B)
& m1_relset_1(F,C,D) )
=> m2_relset_1(k7_relset_1(A,B,C,D,E,F),A,D) ) ).
fof(redefinition_k7_relset_1,axiom,
! [A,B,C,D,E,F] :
( ( m1_relset_1(E,A,B)
& m1_relset_1(F,C,D) )
=> k7_relset_1(A,B,C,D,E,F) = k5_relat_1(E,F) ) ).
fof(dt_k8_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(C,A,B)
=> m2_relset_1(k8_relset_1(A,B,C,D),A,B) ) ).
fof(redefinition_k8_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(C,A,B)
=> k8_relset_1(A,B,C,D) = k7_relat_1(C,D) ) ).
fof(dt_k9_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(D,A,B)
=> m2_relset_1(k9_relset_1(A,B,C,D),A,B) ) ).
fof(redefinition_k9_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(D,A,B)
=> k9_relset_1(A,B,C,D) = k8_relat_1(C,D) ) ).
fof(dt_k10_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(C,A,B)
=> m1_subset_1(k10_relset_1(A,B,C,D),k1_zfmisc_1(B)) ) ).
fof(redefinition_k10_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(C,A,B)
=> k10_relset_1(A,B,C,D) = k9_relat_1(C,D) ) ).
fof(dt_k11_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(C,A,B)
=> m1_subset_1(k11_relset_1(A,B,C,D),k1_zfmisc_1(A)) ) ).
fof(redefinition_k11_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(C,A,B)
=> k11_relset_1(A,B,C,D) = k10_relat_1(C,D) ) ).
%------------------------------------------------------------------------------