Theorem islp2

Description: The predicate " P is a limit point of S " in terms of neighborhoods. Definition of limit point in Munkres p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007) (Proof shortened by Mario Carneiro, 25-Dec-2016)

	Ref	Expression
Hypothesis	lpfval.1	|- X = U. J
Assertion	islp2	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> A. n e. ( ( nei ` J ) ` { P } ) ( n 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		ssdifss	|- ( S C_ X -> ( S \ { P } ) C_ X )
5	1	neindisj2	|- ( ( J e. Top /\ ( S \ { P } ) C_ X /\ P e. X ) -> ( P e. ( ( cls ` J ) ` ( S \ { P } ) ) <-> A. n e. ( ( nei ` J ) ` { P } ) ( n i^i ( S \ { P } ) ) =/= (/) ) )
6	4 5	syl3an2	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( P e. ( ( cls ` J ) ` ( S \ { P } ) ) <-> A. n e. ( ( nei ` J ) ` { P } ) ( n i^i ( S \ { P } ) ) =/= (/) ) )
7	3 6	bitrd	|- ( ( J e. Top /\ S C_ X /\ P e. X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> A. n e. ( ( nei ` J ) ` { P } ) ( n i^i ( S \ { P } ) ) =/= (/) ) )