Theorem islp

Description: The predicate "the class P is a limit point of S ". (Contributed by NM, 10-Feb-2007)

	Ref	Expression
Hypothesis	lpfval.1	|- X = U. J
Assertion	islp	|- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	|- X = U. J
2	1	lpval	|- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )
3	2	eleq2d	|- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> P e. { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } ) )
4		id	|- ( P e. ( ( cls ` J ) ` ( S \ { P } ) ) -> P e. ( ( cls ` J ) ` ( S \ { P } ) ) )
5		id	|- ( x = P -> x = P )
6		sneq	|- ( x = P -> { x } = { P } )
7	6	difeq2d	|- ( x = P -> ( S \ { x } ) = ( S \ { P } ) )
8	7	fveq2d	|- ( x = P -> ( ( cls ` J ) ` ( S \ { x } ) ) = ( ( cls ` J ) ` ( S \ { P } ) ) )
9	5 8	eleq12d	|- ( x = P -> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) ) )
10	4 9	elab3	|- ( P e. { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) )
11	3 10	bitrdi	|- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) ) )