:: SUBSET_1 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

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

registration
let x be set ;
cluster {x} -> non empty ;
coherence
not {x} is empty
by TARSKI:def 1;
let y be set ;
cluster {x,y} -> non empty ;
coherence
not {x,y} is empty
by TARSKI:def 2;
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 -> empty Subset of E equals :: SUBSET_1:def 3
{} ;
coherence
{} is empty 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;

theorem :: SUBSET_1:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being set holds {} is Subset of X
proof end;

theorem :: SUBSET_1:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th7: :: SUBSET_1:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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;
:: 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;

theorem :: SUBSET_1:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th21: :: SUBSET_1:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set holds {} E = ([#] E) ` by XBOOLE_1:37;

theorem :: SUBSET_1:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set holds [#] E = ({} E) ` ;

theorem :: SUBSET_1:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th25: :: SUBSET_1:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A being Subset of E holds A \/ (A ` ) = [#] E
proof end;

theorem Th26: :: SUBSET_1:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A being Subset of E holds A misses A ` by XBOOLE_1:79;

theorem :: SUBSET_1:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A being Subset of E holds A \/ ([#] E) = [#] E
proof end;

theorem :: SUBSET_1:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A, B being Subset of E holds (A \/ B) ` = (A ` ) /\ (B ` ) by XBOOLE_1:53;

theorem :: SUBSET_1:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A, B being Subset of E holds (A /\ B) ` = (A ` ) \/ (B ` ) by XBOOLE_1:54;

theorem Th31: :: SUBSET_1:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A, B being Subset of E holds A \ B = A /\ (B ` )
proof end;

theorem :: SUBSET_1:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A, B being Subset of E holds (A \ B) ` = (A ` ) \/ B
proof end;

theorem :: SUBSET_1:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A, B being Subset of E holds (A \+\ B) ` = (A /\ B) \/ ((A ` ) /\ (B ` ))
proof end;

theorem :: SUBSET_1:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:38  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A being Subset of E holds
( A c= A ` iff A = {} E )
proof end;

theorem :: SUBSET_1:39  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A being Subset of E holds
( A ` c= A iff A = [#] E )
proof end;

theorem :: SUBSET_1:40  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E, X being set
for A being Subset of E st X c= A & X c= A ` holds
X = {}
proof end;

theorem :: SUBSET_1:41  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A, B being Subset of E holds (A \/ B) ` c= A `
proof end;

theorem :: SUBSET_1:42  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E being set
for A, B being Subset of E holds A ` c= (A /\ B) `
proof end;

theorem Th43: :: SUBSET_1:43  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: SUBSET_1:46  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for E, x being set
for A being Subset of E st x in A ` holds
not x in A by XBOOLE_0:def 4;

theorem :: SUBSET_1:55  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being set
for x1 being Element of X st X <> {} holds
{x1} is Subset of X
proof end;

theorem :: SUBSET_1:56  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 , P1[ set ] } :
for X1, X2 being Subset of F1() st ( for y being Element of F1() holds
( y in X1 iff P1[y] ) ) & ( for y being Element of F1() holds
( y in X2 iff P1[y] ) ) holds
X1 = X2
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 ;
assume A1: contradiction ;
func choose S -> Element of S means :: SUBSET_1:def 6
verum;
correctness
existence
ex b1 being Element of S st verum
;
uniqueness
for b1, b2 being Element of S holds b1 = b2
;
by A1;
end;

:: deftheorem defines choose SUBSET_1:def 6 :
for S being set st contradiction holds
for b2 being Element of S holds
( b2 = choose S iff verum );

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;