:: Boolean Properties of Sets - Definitions
:: by Library Committee
::
:: Copyright (c) 2002-2012 Association of Mizar Users

begin

scheme :: XBOOLE_0:sch 1
Separation{ F1() -> set , P1[ set ] } :
ex X being set st
for x being set holds
( x in X iff ( x in F1() & P1[x] ) )
proof end;

definition
let X be set ;
attr X is empty means :Def1: :: XBOOLE_0:def 1
for x being set holds not x in X;
end;

:: deftheorem Def1 defines empty XBOOLE_0:def 1 :
for X being set holds
( X is empty iff for x being set holds not x in X );

registration
cluster empty for set ;
existence
ex b1 being set st b1 is empty
proof end;
end;

definition
func {} -> set equals :: XBOOLE_0:def 2
the empty set ;
coherence
the empty set is set
;
let X, Y be set ;
func X \/ Y -> set means :Def3: :: XBOOLE_0:def 3
for x being set holds
( x in it iff ( x in X or x in Y ) );
existence
ex b1 being set st
for x being set holds
( x in b1 iff ( x in X or x in Y ) )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ( x in X or x in Y ) ) ) & ( for x being set holds
( x in b2 iff ( x in X or x in Y ) ) ) holds
b1 = b2
proof end;
commutativity
for b1, X, Y being set st ( for x being set holds
( x in b1 iff ( x in X or x in Y ) ) ) holds
for x being set holds
( x in b1 iff ( x in Y or x in X ) )
;
idempotence
for X, x being set holds
( x in X iff ( x in X or x in X ) )
;
func X /\ Y -> set means :Def4: :: XBOOLE_0:def 4
for x being set holds
( x in it iff ( x in X & x in Y ) );
existence
ex b1 being set st
for x being set holds
( x in b1 iff ( x in X & x in Y ) )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ( x in X & x in Y ) ) ) & ( for x being set holds
( x in b2 iff ( x in X & x in Y ) ) ) holds
b1 = b2
proof end;
commutativity
for b1, X, Y being set st ( for x being set holds
( x in b1 iff ( x in X & x in Y ) ) ) holds
for x being set holds
( x in b1 iff ( x in Y & x in X ) )
;
idempotence
for X, x being set holds
( x in X iff ( x in X & x in X ) )
;
func X \ Y -> set means :Def5: :: XBOOLE_0:def 5
for x being set holds
( x in it iff ( x in X & not x in Y ) );
existence
ex b1 being set st
for x being set holds
( x in b1 iff ( x in X & not x in Y ) )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ( x in X & not x in Y ) ) ) & ( for x being set holds
( x in b2 iff ( x in X & not x in Y ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines {} XBOOLE_0:def 2 :
{} = the empty set ;

:: deftheorem Def3 defines \/ XBOOLE_0:def 3 :
for X, Y, b3 being set holds
( b3 = X \/ Y iff for x being set holds
( x in b3 iff ( x in X or x in Y ) ) );

:: deftheorem Def4 defines /\ XBOOLE_0:def 4 :
for X, Y, b3 being set holds
( b3 = X /\ Y iff for x being set holds
( x in b3 iff ( x in X & x in Y ) ) );

:: deftheorem Def5 defines \ XBOOLE_0:def 5 :
for X, Y, b3 being set holds
( b3 = X \ Y iff for x being set holds
( x in b3 iff ( x in X & not x in Y ) ) );

definition
let X, Y be set ;
func X \+\ Y -> set equals :: XBOOLE_0:def 6
(X \ Y) \/ (Y \ X);
correctness
coherence
(X \ Y) \/ (Y \ X) is set
;
;
commutativity
for b1, X, Y being set st b1 = (X \ Y) \/ (Y \ X) holds
b1 = (Y \ X) \/ (X \ Y)
;
pred X misses Y means :Def7: :: XBOOLE_0:def 7
X /\ Y = {} ;
symmetry
for X, Y being set st X /\ Y = {} holds
Y /\ X = {}
;
pred X c< Y means :Def8: :: XBOOLE_0:def 8
( X c= Y & X <> Y );
irreflexivity
for X being set holds
( not X c= X or not X <> X )
;
asymmetry
for X, Y being set st X c= Y & X <> Y & Y c= X holds
not Y <> X
proof end;
pred X,Y are_c=-comparable means :: XBOOLE_0:def 9
( X c= Y or Y c= X );
reflexivity
for X being set holds
( X c= X or X c= X )
;
symmetry
for X, Y being set holds
( ( not X c= Y & not Y c= X ) or Y c= X or X c= Y )
;
redefine pred X = Y means :: XBOOLE_0:def 10
( X c= Y & Y c= X );
compatibility
( X = Y iff ( X c= Y & Y c= X ) )
proof end;
end;

:: deftheorem defines \+\ XBOOLE_0:def 6 :
for X, Y being set holds X \+\ Y = (X \ Y) \/ (Y \ X);

:: deftheorem Def7 defines misses XBOOLE_0:def 7 :
for X, Y being set holds
( X misses Y iff X /\ Y = {} );

:: deftheorem Def8 defines c< XBOOLE_0:def 8 :
for X, Y being set holds
( X c< Y iff ( X c= Y & X <> Y ) );

:: deftheorem defines are_c=-comparable XBOOLE_0:def 9 :
for X, Y being set holds
( X,Y are_c=-comparable iff ( X c= Y or Y c= X ) );

:: deftheorem defines = XBOOLE_0:def 10 :
for X, Y being set holds
( X = Y iff ( X c= Y & Y c= X ) );

notation
let X, Y be set ;
antonym X meets Y for X misses Y;
end;

theorem :: XBOOLE_0:1
for x, X, Y being set holds
( x in X \+\ Y iff ( ( x in X & not x in Y ) or ( x in Y & not x in X ) ) )
proof end;

theorem :: XBOOLE_0:2
for X, Y, Z being set st ( for x being set holds
( not x in X iff ( x in Y iff x in Z ) ) ) holds
X = Y \+\ Z
proof end;

registration
coherence
{} is empty
;
end;

registration
let x1 be set ;
cluster {x1} -> non empty ;
coherence
not {x1} is empty
proof end;
let x2 be set ;
cluster {x1,x2} -> non empty ;
coherence
not {x1,x2} is empty
proof end;
end;

registration
cluster non empty for set ;
existence
not for b1 being set holds b1 is empty
proof end;
end;

registration
let D be non empty set ;
let X be set ;
cluster D \/ X -> non empty ;
coherence
not D \/ X is empty
proof end;
cluster X \/ D -> non empty ;
coherence
not X \/ D is empty
;
end;

Lm1: for X being set st X is empty holds
X = {}

proof end;

theorem Th3: :: XBOOLE_0:3
for X, Y being set holds
( X meets Y iff ex x being set st
( x in X & x in Y ) )
proof end;

theorem :: XBOOLE_0:4
for X, Y being set holds
( X meets Y iff ex x being set st x in X /\ Y )
proof end;

theorem :: XBOOLE_0:5
for X, Y, x being set st X misses Y & x in X \/ Y & not ( x in X & not x in Y ) holds
( x in Y & not x in X ) by ;

scheme :: XBOOLE_0:sch 2
Extensionality{ F1() -> set , F2() -> set , P1[ set ] } :
F1() = F2()
provided
A1: for x being set holds
( x in F1() iff P1[x] ) and
A2: for x being set holds
( x in F2() iff P1[x] )
proof end;

scheme :: XBOOLE_0:sch 3
SetEq{ P1[ set ] } :
for X1, X2 being set st ( for x being set holds
( x in X1 iff P1[x] ) ) & ( for x being set holds
( x in X2 iff P1[x] ) ) holds
X1 = X2
proof end;

begin

:: from RLSUB_2, 2006.12.02, AT
theorem :: XBOOLE_0:6
for X, Y being set st X c< Y holds
ex x being set st
( x in Y & not x in X )
proof end;

:: 2008.08.07, A.T.
theorem :: XBOOLE_0:7
for X being set st X <> {} holds
ex x being set st x in X
proof end;