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Metamath Proof Explorer

Theorem lpsscls

Description: The limit points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007)

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

Proof

Step Hyp Ref Expression
1 lpfval.1 
 |-  X = U. J
2 1 lpval 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )
3 difss 
 |-  ( S \ { x } ) C_ S
4 1 clsss 
 |-  ( ( J e. Top /\ S C_ X /\ ( S \ { x } ) C_ S ) -> ( ( cls ` J ) ` ( S \ { x } ) ) C_ ( ( cls ` J ) ` S ) )
5 3 4 mp3an3 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` ( S \ { x } ) ) C_ ( ( cls ` J ) ` S ) )
6 5 sseld 
 |-  ( ( J e. Top /\ S C_ X ) -> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) -> x e. ( ( cls ` J ) ` S ) ) )
7 6 abssdv 
 |-  ( ( J e. Top /\ S C_ X ) -> { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } C_ ( ( cls ` J ) ` S ) )
8 2 7 eqsstrd 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) C_ ( ( cls ` J ) ` S ) )