This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )