Theorem islp3

Description: The predicate " P is a limit point of S " in terms of open sets. see islp2 , elcls , islp . (Contributed by FL, 31-Jul-2009)

	Ref	Expression
Hypothesis	lpfval.1	|- X = U. J
Assertion	islp3	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> A. x e. J ( P e. x -> ( x i^i ( S \ { P } ) ) =/= (/) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	|- X = U. J
2	1	islp	|- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) ) )
3	2	3adant3	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) ) )
4		simp2	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> S C_ X )
5	4	ssdifssd	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( S \ { P } ) C_ X )
6	1	elcls	|- ( ( J e. Top /\ ( S \ { P } ) C_ X /\ P e. X ) -> ( P e. ( ( cls ` J ) ` ( S \ { P } ) ) <-> A. x e. J ( P e. x -> ( x i^i ( S \ { P } ) ) =/= (/) ) ) )
7	5 6	syld3an2	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( P e. ( ( cls ` J ) ` ( S \ { P } ) ) <-> A. x e. J ( P e. x -> ( x i^i ( S \ { P } ) ) =/= (/) ) ) )
8	3 7	bitrd	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> A. x e. J ( P e. x -> ( x i^i ( S \ { P } ) ) =/= (/) ) ) )