Theorem islpi

Description: A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007)

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

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	|- X = U. J
2	1	clslp	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( S u. ( ( limPt ` J ) ` S ) ) )
3	2	eleq2d	|- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( cls ` J ) ` S ) <-> P e. ( S u. ( ( limPt ` J ) ` S ) ) ) )
4		elun	|- ( P e. ( S u. ( ( limPt ` J ) ` S ) ) <-> ( P e. S \/ P e. ( ( limPt ` J ) ` S ) ) )
5		df-or	|- ( ( P e. S \/ P e. ( ( limPt ` J ) ` S ) ) <-> ( -. P e. S -> P e. ( ( limPt ` J ) ` S ) ) )
6	4 5	bitri	|- ( P e. ( S u. ( ( limPt ` J ) ` S ) ) <-> ( -. P e. S -> P e. ( ( limPt ` J ) ` S ) ) )
7	3 6	bitrdi	|- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( cls ` J ) ` S ) <-> ( -. P e. S -> P e. ( ( limPt ` J ) ` S ) ) ) )
8	7	biimpd	|- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( cls ` J ) ` S ) -> ( -. P e. S -> P e. ( ( limPt ` J ) ` S ) ) ) )
9	8	imp32	|- ( ( ( J e. Top /\ S C_ X ) /\ ( P e. ( ( cls ` J ) ` S ) /\ -. P e. S ) ) -> P e. ( ( limPt ` J ) ` S ) )