:: Properties of Subsets
:: by Zinaida Trybulec
::
:: Received March 4, 1989
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

registration

let X be set ;
cluster bool X -> non empty ;
coherence
not bool X is empty by ZFMISC_1:def 1;

end;

registration

let x1, x2, x3 be set ;
cluster {x1,x2,x3} -> non empty ;
coherence
not {x1,x2,x3} is empty by ENUMSET1:def 1;
let x4 be set ;
cluster {x1,x2,x3,x4} -> non empty ;
coherence
not {x1,x2,x3,x4} is empty by ENUMSET1:def 2;
let x5 be set ;
cluster {x1,x2,x3,x4,x5} -> non empty ;
coherence
not {x1,x2,x3,x4,x5} is empty by ENUMSET1:def 3;
let x6 be set ;
cluster {x1,x2,x3,x4,x5,x6} -> non empty ;
coherence
not {x1,x2,x3,x4,x5,x6} is empty by ENUMSET1:def 4;
let x7 be set ;
cluster {x1,x2,x3,x4,x5,x6,x7} -> non empty ;
coherence
not {x1,x2,x3,x4,x5,x6,x7} is empty by ENUMSET1:def 5;
let x8 be set ;
cluster {x1,x2,x3,x4,x5,x6,x7,x8} -> non empty ;
coherence
not {x1,x2,x3,x4,x5,x6,x7,x8} is empty by ENUMSET1:def 6;
let x9 be set ;
cluster {x1,x2,x3,x4,x5,x6,x7,x8,x9} -> non empty ;
coherence
not {x1,x2,x3,x4,x5,x6,x7,x8,x9} is empty by ENUMSET1:def 7;
let x10 be set ;
cluster {x1,x2,x3,x4,x5,x6,x7,x8,x9,x10} -> non empty ;
coherence
not {x1,x2,x3,x4,x5,x6,x7,x8,x9,x10} is empty by ENUMSET1:def 8;

end;

definition

let X be set ;
canceled;
mode Element of X -> set means :Def2: :: SUBSET_1:def 2
it in X if not X is empty
otherwise it is empty ;
existence
( ( not X is empty implies ex b₁ being set st b₁ in X ) & ( X is empty implies ex b₁ being set st b₁ is empty ) ) by XBOOLE_0:def 1;
consistency
for b₁ being set holds verum ;

end;

:: deftheorem SUBSET_1:def 1 :
canceled;

:: deftheorem Def2 defines Element SUBSET_1:def 2 :
for X being set
for b₂ being set holds
( ( not X is empty implies ( b₂ is Element of X iff b₂ in X ) ) & ( X is empty implies ( b₂ is Element of X iff b₂ is empty ) ) );

definition

let X be set ;
mode Subset of X is Element of bool X;

end;

registration

let X be non empty set ;
cluster non empty Element of bool X;
existence
not for b₁ being Subset of X holds b₁ is empty

proof end;

end;

registration

let X1, X2 be non empty set ;
cluster [:X1,X2:] -> non empty ;
coherence
not [:X1,X2:] is empty

proof end;

end;

registration

let X1, X2, X3 be non empty set ;
cluster [:X1,X2,X3:] -> non empty ;
coherence
not [:X1,X2,X3:] is empty

proof end;

end;

registration

let X1, X2, X3, X4 be non empty set ;
cluster [:X1,X2,X3,X4:] -> non empty ;
coherence
not [:X1,X2,X3,X4:] is empty

proof end;

end;

definition

let D be non empty set ;
let X be non empty Subset of D;
:: original: Element
redefine mode Element of X -> Element of D;
coherence
for b₁ being Element of X holds b₁ is Element of D

proof end;

end;

Lm1: for E being set
for X being Subset of E
for x being set st x in X holds
x in E

proof end;

registration

let E be set ;
cluster empty Element of bool E;
existence
ex b₁ being Subset of E st b₁ is empty

proof end;

end;

definition

let E be set ;
func {} E -> Subset of E equals :: SUBSET_1:def 3
{} ;
coherence
{} is Subset of E

proof end;

correctness
;
func [#] E -> Subset of E equals :: SUBSET_1:def 4
E;
coherence
E is Subset of E

proof end;

correctness
;

end;

:: deftheorem defines {} SUBSET_1:def 3 :
for E being set holds {} E = {} ;

:: deftheorem defines [#] SUBSET_1:def 4 :
for E being set holds [#] E = E;

registration

let E be set ;
cluster {} E -> empty ;
coherence
{} E is empty ;

end;

theorem :: SUBSET_1:1

canceled;

theorem :: SUBSET_1:2

canceled;

theorem :: SUBSET_1:3

canceled;

theorem :: SUBSET_1:4

for X being set holds {} is Subset of X

proof end;

theorem :: SUBSET_1:5

canceled;

theorem :: SUBSET_1:6

canceled;

theorem Th7: :: SUBSET_1:7

for E being set
for A, B being Subset of E st ( for x being Element of E st x in A holds
x in B ) holds
A c= B

proof end;

theorem Th8: :: SUBSET_1:8

for E being set
for A, B being Subset of E st ( for x being Element of E holds
( x in A iff x in B ) ) holds
A = B

proof end;

theorem :: SUBSET_1:9

canceled;

theorem :: SUBSET_1:10

for E being set
for A being Subset of E st A <> {} holds
ex x being Element of E st x in A

proof end;

definition

let E be set ;
let A be Subset of E;
func A ` -> Subset of E equals :: SUBSET_1:def 5
E \ A;
coherence
E \ A is Subset of E

proof end;

correctness
;
involutiveness
for b₁, b₂ being Subset of E st b₁ = E \ b₂ holds
b₂ = E \ b₁

proof end;

let B be Subset of E;
:: original: \/
redefine func A \/ B -> Subset of E;
coherence
A \/ B is Subset of E

proof end;

:: original: \+\
redefine func A \+\ B -> Subset of E;
coherence
A \+\ B is Subset of E

proof end;

end;

:: deftheorem defines ` SUBSET_1:def 5 :
for E being set
for A being Subset of E holds A ` = E \ A;

definition

let X, Y be set ;
:: original: \
redefine func X \ Y -> Subset of X;
coherence
X \ Y is Subset of X

proof end;

end;

definition

let E be set ;
let A be Subset of E;
let X be set ;
:: original: \
redefine func A \ X -> Subset of E;
coherence
A \ X is Subset of E

proof end;

end;

definition

let E be set ;
let A be Subset of E;
let X be set ;
:: original: /\
redefine func A /\ X -> Subset of E;
coherence
A /\ X is Subset of E

proof end;

end;

definition

let E, X be set ;
let A be Subset of E;
:: original: /\
redefine func X /\ A -> Subset of E;
coherence
X /\ A is Subset of E

proof end;

end;

theorem :: SUBSET_1:11

canceled;

theorem :: SUBSET_1:12

canceled;

theorem :: SUBSET_1:13

canceled;

theorem :: SUBSET_1:14

canceled;

theorem :: SUBSET_1:15

for E being set
for A, B, C being Subset of E st ( for x being Element of E holds
( x in A iff ( x in B or x in C ) ) ) holds
A = B \/ C

proof end;

theorem :: SUBSET_1:16

for E being set
for A, B, C being Subset of E st ( for x being Element of E holds
( x in A iff ( x in B & x in C ) ) ) holds
A = B /\ C

proof end;

theorem :: SUBSET_1:17

for E being set
for A, B, C being Subset of E st ( for x being Element of E holds
( x in A iff ( x in B & not x in C ) ) ) holds
A = B \ C

proof end;

theorem :: SUBSET_1:18

for E being set
for A, B, C being Subset of E st ( for x being Element of E holds
( x in A iff ( ( x in B & not x in C ) or ( x in C & not x in B ) ) ) ) holds
A = B \+\ C

proof end;

theorem :: SUBSET_1:19

canceled;

theorem :: SUBSET_1:20

canceled;

theorem :: SUBSET_1:21

canceled;

theorem :: SUBSET_1:22

for E being set holds [#] E = ({} E) ` ;

theorem :: SUBSET_1:23

canceled;

theorem :: SUBSET_1:24

canceled;

theorem Th25: :: SUBSET_1:25

for E being set
for A being Subset of E holds A \/ (A `) = [#] E

proof end;

theorem :: SUBSET_1:26

canceled;

theorem :: SUBSET_1:27

canceled;

theorem :: SUBSET_1:28

for E being set
for A being Subset of E holds A \/ ([#] E) = [#] E

proof end;

theorem :: SUBSET_1:29

canceled;

theorem :: SUBSET_1:30

canceled;

theorem Th31: :: SUBSET_1:31

for E being set
for A, B being Subset of E holds
( A c= B iff B ` c= A ` )

proof end;

theorem :: SUBSET_1:32

for E being set
for A, B being Subset of E holds A \ B = A /\ (B `)

proof end;

theorem :: SUBSET_1:33

for E being set
for A, B being Subset of E holds (A \ B) ` = (A `) \/ B

proof end;

theorem :: SUBSET_1:34

for E being set
for A, B being Subset of E holds (A \+\ B) ` = (A /\ B) \/ ((A `) /\ (B `))

proof end;

theorem :: SUBSET_1:35

for E being set
for A, B being Subset of E st A c= B ` holds
B c= A `

proof end;

theorem :: SUBSET_1:36

for E being set
for A, B being Subset of E st A ` c= B holds
B ` c= A

proof end;

theorem :: SUBSET_1:37

canceled;

theorem :: SUBSET_1:38

for E being set
for A being Subset of E holds
( A c= A ` iff A = {} E )

proof end;

theorem :: SUBSET_1:39

for E being set
for A being Subset of E holds
( A ` c= A iff A = [#] E )

proof end;

theorem :: SUBSET_1:40

for E, X being set
for A being Subset of E st X c= A & X c= A ` holds
X = {} by XBOOLE_1:67, XBOOLE_1:79;

theorem :: SUBSET_1:41

for E being set
for A, B being Subset of E holds (A \/ B) ` c= A `

proof end;

theorem :: SUBSET_1:42

for E being set
for A, B being Subset of E holds A ` c= (A /\ B) `

proof end;

theorem Th43: :: SUBSET_1:43

for E being set
for A, B being Subset of E holds
( A misses B iff A c= B ` )

proof end;

theorem :: SUBSET_1:44

for E being set
for A, B being Subset of E holds
( A misses B ` iff A c= B )

proof end;

theorem :: SUBSET_1:45

canceled;

theorem :: SUBSET_1:46

for E being set
for A, B being Subset of E st A misses B & A ` misses B ` holds
A = B `

proof end;

theorem :: SUBSET_1:47

for E being set
for A, B, C being Subset of E st A c= B & C misses B holds
A c= C `

proof end;

theorem :: SUBSET_1:48

for E being set
for A, B being Subset of E st ( for a being Element of A holds a in B ) holds
A c= B

proof end;

theorem :: SUBSET_1:49

for E being set
for A being Subset of E st ( for x being Element of E holds x in A ) holds
E = A

proof end;

theorem :: SUBSET_1:50

for E being set st E <> {} holds
for B being Subset of E
for x being Element of E st not x in B holds
x in B `

proof end;

theorem Th51: :: SUBSET_1:51

for E being set
for A, B being Subset of E st ( for x being Element of E holds
( x in A iff not x in B ) ) holds
A = B `

proof end;

theorem :: SUBSET_1:52

for E being set
for A, B being Subset of E st ( for x being Element of E holds
( not x in A iff x in B ) ) holds
A = B `

proof end;

theorem :: SUBSET_1:53

for E being set
for A, B being Subset of E st ( for x being Element of E holds
( ( x in A & not x in B ) or ( x in B & not x in A ) ) ) holds
A = B `

proof end;

theorem :: SUBSET_1:54

canceled;

theorem :: SUBSET_1:55

for X being set
for x1 being Element of X st X <> {} holds
{x1} is Subset of X

proof end;

theorem :: SUBSET_1:56

for X being set
for x1, x2 being Element of X st X <> {} holds
{x1,x2} is Subset of X

proof end;

theorem :: SUBSET_1:57

for X being set
for x1, x2, x3 being Element of X st X <> {} holds
{x1,x2,x3} is Subset of X

proof end;

theorem :: SUBSET_1:58

for X being set
for x1, x2, x3, x4 being Element of X st X <> {} holds
{x1,x2,x3,x4} is Subset of X

proof end;

theorem :: SUBSET_1:59

for X being set
for x1, x2, x3, x4, x5 being Element of X st X <> {} holds
{x1,x2,x3,x4,x5} is Subset of X

proof end;

theorem :: SUBSET_1:60

for X being set
for x1, x2, x3, x4, x5, x6 being Element of X st X <> {} holds
{x1,x2,x3,x4,x5,x6} is Subset of X

proof end;

theorem :: SUBSET_1:61

for X being set
for x1, x2, x3, x4, x5, x6, x7 being Element of X st X <> {} holds
{x1,x2,x3,x4,x5,x6,x7} is Subset of X

proof end;

theorem :: SUBSET_1:62

for X being set
for x1, x2, x3, x4, x5, x6, x7, x8 being Element of X st X <> {} holds
{x1,x2,x3,x4,x5,x6,x7,x8} is Subset of X

proof end;

theorem :: SUBSET_1:63

for x, X being set st x in X holds
{x} is Subset of X

proof end;

scheme :: SUBSET_1:sch 1
SubsetEx{ F₁() -> set , P₁[ set ] } :

ex X being Subset of F₁() st
for x being set holds
( x in X iff ( x in F₁() & P₁[x] ) )

proof end;

scheme :: SUBSET_1:sch 2
SubsetEq{ F₁() -> set , F₂() -> Subset of F₁(), F₃() -> Subset of F₁(), P₁[ set ] } :

F₂() = F₃()

provided

A1: for y being Element of F₁() holds
( y in F₂() iff P₁[y] ) and
A2: for y being Element of F₁() holds
( y in F₃() iff P₁[y] )

proof end;

definition

let X, Y be non empty set ;
:: original: misses
redefine pred X misses Y;
irreflexivity
for X being non empty set holds not X misses X

proof end;

end;

definition

let X, Y be non empty set ;
:: original: misses
redefine pred X meets Y;
reflexivity
for X being non empty set holds not X misses X

proof end;

end;

definition

let S be set ;
func choose S -> Element of S equals :: SUBSET_1:def 6
the Element of S;
correctness
coherence
the Element of S is Element of S;
;

end;

:: deftheorem defines choose SUBSET_1:def 6 :
for S being set holds choose S = the Element of S;

begin

Lm2: for X, Y being set st ( for x being set st x in X holds
x in Y ) holds
X is Subset of Y

proof end;

Lm3: for x, E being set
for A being Subset of E st x in A holds
x is Element of E

proof end;

scheme :: SUBSET_1:sch 3
SubsetEx{ F₁() -> non empty set , P₁[ set ] } :

ex B being Subset of F₁() st
for x being Element of F₁() holds
( x in B iff P₁[x] )

proof end;

scheme :: SUBSET_1:sch 4
SubComp{ F₁() -> set , F₂() -> Subset of F₁(), F₃() -> Subset of F₁(), P₁[ set ] } :

F₂() = F₃()

provided

A1: for X being Element of F₁() holds
( X in F₂() iff P₁[X] ) and
A2: for X being Element of F₁() holds
( X in F₃() iff P₁[X] )

proof end;

theorem :: SUBSET_1:64

for E being set
for A, B being Subset of E st A ` = B ` holds
A = B

proof end;

registration

let X be empty set ;
cluster -> empty Element of bool X;
coherence
for b₁ being Subset of X holds b₁ is empty

proof end;

end;

definition

let E be set ;
let A be Subset of E;
attr A is proper means :: SUBSET_1:def 7
A <> E;

end;

:: deftheorem defines proper SUBSET_1:def 7 :
for E being set
for A being Subset of E holds
( A is proper iff A <> E );

registration

let E be set ;
cluster [#] E -> non proper ;
coherence
not [#] E is proper

proof end;

end;

registration

let E be set ;
cluster non proper Element of bool E;
existence
not for b₁ being Subset of E holds b₁ is proper

proof end;

end;

registration

let E be non empty set ;
cluster non proper -> non empty Element of bool E;
coherence
for b₁ being Subset of E st not b₁ is proper holds
not b₁ is empty

proof end;

cluster empty -> proper Element of bool E;
coherence
for b₁ being Subset of E st b₁ is empty holds
b₁ is proper ;

end;

registration

let E be non empty set ;
cluster proper Element of bool E;
existence
ex b₁ being Subset of E st b₁ is proper

proof end;

end;

registration

let E be empty set ;
cluster -> non proper Element of bool E;
coherence
for b₁ being Subset of E holds not b₁ is proper

proof end;

end;

theorem :: SUBSET_1:65

for X, Y, A, z being set st z in A & A c= [:X,Y:] holds
ex x being Element of X ex y being Element of Y st z = [x,y]

proof end;