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Metamath Proof Explorer

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 } ) ) =/= (/) ) )