:: Properties of Subsets
:: by Zinaida Trybulec
::
:: Received March 4, 1989
:: Copyright (c) 1990-2015 Association of Mizar Users


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 object ;
cluster {x1,x2,x3} -> non empty ;
coherence
not {x1,x2,x3} is empty by ENUMSET1:def 1;
let x4 be object ;
cluster {x1,x2,x3,x4} -> non empty ;
coherence
not {x1,x2,x3,x4} is empty by ENUMSET1:def 2;
let x5 be object ;
cluster {x1,x2,x3,x4,x5} -> non empty ;
coherence
not {x1,x2,x3,x4,x5} is empty by ENUMSET1:def 3;
let x6 be object ;
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 object ;
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 object ;
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 object ;
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 object ;
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 ;
mode Element of X -> set means :Def1: :: SUBSET_1:def 1
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 ) )
proof end;
consistency
for b1 being set holds verum ;
sethood
( ( not X is empty & ex b1 being set st
for b2 being set st b2 in X holds
b2 in b1 ) or ( X is empty & ex b1 being set st
for b2 being set st b2 is empty holds
b2 in b1 ) )
proof end;
end;

:: deftheorem Def1 defines Element SUBSET_1:def 1 :
for X, 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 for 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 ;
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 ;
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 object st x in X holds
x in E
proof end;

registration
let E be set ;
cluster empty for 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 2
 {} ;
coherence
{} is Subset of E
proof end;
correctness
;
func [#] E -> Subset of E equals :: SUBSET_1:def 3
E;
coherence
E is Subset of E
proof end;
correctness
;
end;

:: deftheorem defines {} SUBSET_1:def 2 :
for E being set holds {} E = {} ;

:: deftheorem defines [#] SUBSET_1:def 3 :
for E being set holds [#] E = E;

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

theorem :: SUBSET_1:1
for X being set holds {} is Subset of X
proof end;

theorem Th2: :: SUBSET_1:2
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 :: SUBSET_1:3
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 by Th2;

theorem Th4: :: SUBSET_1:4
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 4
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 4 :
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_1:5
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_1:6
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_1:7
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_1:8
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_1:9
for E being set holds [#] E = ({} E) ` ;

theorem Th10: :: SUBSET_1:10
for E being set
for A being Subset of E holds A \/ (A `) = [#] E
proof end;

theorem :: SUBSET_1:11
for E being set
for A being Subset of E holds A \/ ([#] E) = [#] E
proof end;

theorem Th12: :: SUBSET_1:12
for E being set
for A, B being Subset of E holds
( A c= B iff B ` c= A ` )
proof end;

theorem :: SUBSET_1:13
for E being set
for A, B being Subset of E holds A \ B = A /\ (B `)
proof end;

theorem :: SUBSET_1:14
for E being set
for A, B being Subset of E holds (A \ B) ` = (A `) \/ B
proof end;

theorem :: SUBSET_1:15
for E being set
for A, B being Subset of E holds (A \+\ B) ` = (A /\ B) \/ ((A `) /\ (B `))
proof end;

theorem :: SUBSET_1:16
for E being set
for A, B being Subset of E st A c= B ` holds
B c= A `
proof end;

theorem :: SUBSET_1:17
for E being set
for A, B being Subset of E st A ` c= B holds
B ` c= A
proof end;

theorem :: SUBSET_1:18
for E being set
for A being Subset of E holds
( A c= A ` iff A = {} E ) by XBOOLE_1:38;

theorem :: SUBSET_1:19
for E being set
for A being Subset of E holds
( A ` c= A iff A = [#] E )
proof end;

theorem :: SUBSET_1:20
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_1:21
for E being set
for A, B being Subset of E holds (A \/ B) ` c= A ` by Th12, XBOOLE_1:7;

theorem :: SUBSET_1:22
for E being set
for A, B being Subset of E holds A ` c= (A /\ B) ` by Th12, XBOOLE_1:17;

theorem Th23: :: SUBSET_1:23
for E being set
for A, B being Subset of E holds
( A misses B iff A c= B ` )
proof end;

theorem :: SUBSET_1:24
for E being set
for A, B being Subset of E holds
( A misses B ` iff A c= B )
proof end;

theorem :: SUBSET_1:25
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_1:26
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;

::
:: ADDITIONAL THEOREMS
::
theorem :: SUBSET_1:27
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_1:28
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_1:29
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 Th30: :: SUBSET_1:30
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_1:31
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_1:32
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_1:33
for X being set
for x1 being Element of X st X <> {} holds
{x1} is Subset of X
proof end;

theorem :: SUBSET_1:34
for X being set
for x1, x2 being Element of X st X <> {} holds
{x1,x2} is Subset of X
proof end;

theorem :: SUBSET_1:35
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_1:36
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_1:37
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_1:38
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_1:39
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_1:40
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_1:41
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[ object ] } :
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 R6(b1,b1) ;
end;

definition
let X, Y be non empty set ;
:: original: misses
redefine pred X meets Y;
reflexivity
for X being non empty set holds R6(b1,b1) ;
end;

definition
end;

:: deftheorem SUBSET_1:def 5 :
canceled;

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

Lm3: for E being set
for A being Subset of E
for x being object st x in A holds
x is Element of E
proof end;

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

:: from FIN_TOPO, 2006.11.07. A.T.
theorem :: SUBSET_1:42
for E being set
for A, B being Subset of E st A ` = B ` holds
A = B
proof end;

:: from SGRAPH1, 2008.06.02
registration
let X be empty set ;
cluster -> empty for Element of bool X;
coherence
for b1 being Subset of X holds b1 is empty
proof end;
end;

:: from TEX_2, 2006.06.18, A.T. (simplified)
definition
let E be set ;
let A be Subset of E;
attr A is proper means :: SUBSET_1:def 6
A <> E;
end;

:: deftheorem defines proper SUBSET_1:def 6 :
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 ;
end;

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

:: from GRCAT_1, 2010.02.18, A.T.
theorem :: SUBSET_1:43
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;

:: from BORSUK_4, 2011.03.04, A.T.
theorem :: SUBSET_1:44
for X being non empty set
for A, B being non empty Subset of X st A c< B holds
ex p being Element of X st
( p in B & A c= B \ {p} )
proof end;

definition
let X be set ;
redefine attr X is trivial means :: SUBSET_1:def 7
 for x, y being Element of X holds x = y;
compatibility
( X is trivial iff for x, y being Element of X holds x = y )
proof end;
end;

:: deftheorem defines trivial SUBSET_1:def 7 :
for X being set holds
( X is trivial iff for x, y being Element of X holds x = y );

registration
let X be non empty set ;
cluster non empty trivial for Element of bool X;
existence
ex b1 being Subset of X st
( not b1 is empty & b1 is trivial )
proof end;
end;

registration
let X be trivial set ;
cluster -> trivial for Element of bool X;
coherence
for b1 being Subset of X holds b1 is trivial
proof end;
end;

registration
let X be non trivial set ;
cluster non trivial for Element of bool X;
existence
not for b1 being Subset of X holds b1 is trivial
proof end;
end;

theorem :: SUBSET_1:45
for D being set
for A being Subset of D st not A is trivial holds
ex d1, d2 being Element of D st
( d1 in A & d2 in A & d1 <> d2 )
proof end;

theorem Th46: :: SUBSET_1:46
for X being non empty trivial set ex x being Element of X st X = {x}
proof end;

:: from BORSUK_4, 2011.07.31, A.T.
theorem :: SUBSET_1:47
for X being non empty set
for A being non empty Subset of X st A is trivial holds
ex x being Element of X st A = {x}
proof end;

theorem :: SUBSET_1:48
for X being non trivial set
for x being Element of X ex y being object st
( y in X & x <> y )
proof end;

definition
let x be object ;
let X be set ;
assume A1: x in X ;
func In (x,X) -> Element of X equals :Def7: :: SUBSET_1:def 8
x;
correctness
coherence
x is Element of X;
proof end;
end;

:: deftheorem Def7 defines In SUBSET_1:def 8 :
for x being object
for X being set st x in X holds
In (x,X) = x;

theorem :: SUBSET_1:49
for X, Y being set
for x being object st x in X /\ Y holds
In (x,X) = In (x,Y)
proof end;

:: from BORSUK_4, 2012.08.12, A.T.
theorem :: SUBSET_1:50
for X being non trivial set
for p being set ex q being Element of X st q <> p
proof end;

theorem :: SUBSET_1:51
for T being non trivial set
for X being non trivial Subset of T
for p being set ex q being Element of T st
( q in X & q <> p )
proof end;

:: 2012.12.16, A.T.
scheme :: SUBSET_1:sch 5
Union1{ F1() -> set , F2() -> Element of F1(), F3( object ) -> set } :
union { F3(j) where j is Element of F1() : j in {F2()} } = F3(F2())
proof end;

scheme :: SUBSET_1:sch 6
Union2{ F1() -> set , F2() -> Element of F1(), F3() -> Element of F1(), F4( object ) -> set } :
union { F4(j) where j is Element of F1() : j in {F2(),F3()} } = F4(F2()) \/ F4(F3())
proof end;