XBOOLE_0: Boolean Properties of Sets

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 :

registration

cluster empty for set ;

existence

proof end;

end;

definition

func {} -> set equals :: XBOOLE_0:def 2
the empty set ;

coherence ;
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

proof end;

uniqueness

proof end;

commutativity ;
idempotence ;

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

proof end;

uniqueness

proof end;

commutativity ;
idempotence ;

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

proof end;

uniqueness

proof end;

end;

:: deftheorem defines {} XBOOLE_0:def 2 :

:: deftheorem Def3 defines \/ XBOOLE_0:def 3 :

:: deftheorem Def4 defines /\ XBOOLE_0:def 4 :

:: deftheorem Def5 defines \ XBOOLE_0:def 5 :

definition

let X, Y be set ;

func X \+\ Y -> set equals :: XBOOLE_0:def 6
(X \ Y) \/ (Y \ X);

correctness ;
commutativity ;

pred X misses Y means :Def7: :: XBOOLE_0:def 7
X /\ Y = {} ;

symmetry ;

pred X c< Y means :Def8: :: XBOOLE_0:def 8
( X c= Y & X <> Y );

irreflexivity ;
asymmetry

proof end;

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 );

compatibility

proof end;

end;

:: deftheorem defines \+\ XBOOLE_0:def 6 :

:: deftheorem Def7 defines misses XBOOLE_0:def 7 :

:: deftheorem Def8 defines c< XBOOLE_0:def 8 :

:: deftheorem defines are_c=-comparable XBOOLE_0:def 9 :

:: deftheorem defines = XBOOLE_0:def 10 :

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

cluster {} -> empty ;

coherence ;

end;

registration

let x1 be set ;

cluster {x1} -> non empty ;

coherence

proof end;

let x2 be set ;

cluster {x1,x2} -> non empty ;

coherence

proof end;

end;

registration

cluster non empty for set ;

existence

proof end;

end;

registration

let D be non empty set ;
let X be set ;

cluster D \/ X -> non empty ;

coherence

proof end;

cluster X \/ D -> non empty ;

coherence ;

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 Def3, Th3;

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;

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;

theorem :: XBOOLE_0:7

for X being set st X <> {} holds
ex x being set st x in X

proof end;