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Metamath Proof Explorer

Theorem flimnei

Description: A filter contains all of the neighborhoods of its limit points. (Contributed by Jeff Hankins, 4-Sep-2009) (Revised by Mario Carneiro, 9-Apr-2015)

Ref Expression
Assertion flimnei ⊢ A ∈ J fLim F ∧ N ∈ nei ⁡ J ⁡ A → N ∈ F

Proof

Step Hyp Ref Expression
1 flimneiss ⊢ A ∈ J fLim F → nei ⁡ J ⁡ A ⊆ F
2 1 sselda ⊢ A ∈ J fLim F ∧ N ∈ nei ⁡ J ⁡ A → N ∈ F