:: Boolean Properties of Sets - Definitions
:: by Library Committee
::
:: Received April 6, 2002
:: Copyright (c) 2002-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies TARSKI, XBOOLE_0, MATROID0, AOFA_000;
notations TARSKI;
constructors TARSKI;
begin
reserve X, Y, Z for set, x, y, z for object;
scheme :: XBOOLE_0:sch 1
Separation { A()-> set, P[object] } :
ex X being set st for x holds x in X iff x in A() & P[x];
definition
let X be set;
attr X is empty means
:: XBOOLE_0:def 1
not ex x st x in X;
end;
registration
cluster empty for set;
end;
definition
func {} -> set equals
:: XBOOLE_0:def 2
the empty set;
let X,Y be set;
func X \/ Y -> set means
:: XBOOLE_0:def 3
for x holds x in it iff x in X or x in Y;
commutativity;
idempotence;
func X /\ Y -> set means
:: XBOOLE_0:def 4
for x holds x in it iff x in X & x in Y;
commutativity;
idempotence;
func X \ Y -> set means
:: XBOOLE_0:def 5
for x holds x in it iff x in X & not x in Y;
end;
definition
let X, Y be set;
func X \+\ Y -> set equals
:: XBOOLE_0:def 6
(X \ Y) \/ (Y \ X);
commutativity;
pred X misses Y means
:: XBOOLE_0:def 7
X /\ Y = {};
symmetry;
pred X c< Y means
:: XBOOLE_0:def 8
X c= Y & X <> Y;
irreflexivity;
asymmetry;
pred X,Y are_c=-comparable means
:: XBOOLE_0:def 9
X c= Y or Y c= X;
reflexivity;
symmetry;
redefine pred X = Y means
:: XBOOLE_0:def 10
X c= Y & Y c= X;
end;
notation
let X, Y be set;
antonym X meets Y for X misses Y;
end;
theorem :: XBOOLE_0:1
x in X \+\ Y iff not (x in X iff x in Y);
theorem :: XBOOLE_0:2
(for x holds not x in X iff (x in Y iff x in Z)) implies X = Y \+\ Z;
registration
cluster {} -> empty;
end;
registration
let x;
cluster { x } -> non empty;
let y;
cluster { x, y } -> non empty;
end;
registration
cluster non empty for set;
end;
registration
let D be non empty set, X be set;
cluster D \/ X -> non empty;
cluster X \/ D -> non empty;
end;
theorem :: XBOOLE_0:3
X meets Y iff ex x st x in X & x in Y;
theorem :: XBOOLE_0:4
X meets Y iff ex x st x in X /\ Y;
theorem :: XBOOLE_0:5
X misses Y & x in X \/ Y implies x in X & not x in Y or x in Y & not x in X;
scheme :: XBOOLE_0:sch 2
Extensionality { X,Y() -> set, P[object] } :
X() = Y()
provided
for x holds x in X() iff P[x] and
for x holds x in Y() iff P[x];
scheme :: XBOOLE_0:sch 3
SetEq { P[object] } :
for X1,X2 being set st
(for x holds x in X1 iff P[x]) &
(for x holds x in X2 iff P[x]) holds X1 = X2;
begin :: Addenda
:: from RLSUB_2, 2006.12.02, AT
theorem :: XBOOLE_0:6
X c< Y implies ex x st x in Y & not x in X;
:: 2008.08.07, A.T.
theorem :: XBOOLE_0:7
X <> {} implies ex x st x in X;
:: 2012.10.08, A.T.
theorem :: XBOOLE_0:8
X c< Y implies ex x st x in Y & X c= Y \ {x};
:: from MATROID0, 2013.01.18, A.T.
notation
let x,y be set;
antonym x c/= y for x c= y;
end;
:: from AOFA_000, 2013.01.18, A.T.
notation
let x be object,y be set;
antonym x nin y for x in y;
end;