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Metamath Proof Explorer

Theorem fclstop

Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015) (Revised by Stefan O'Rear, 8-Aug-2015)

Ref Expression
Assertion fclstop ⊢ A ∈ J fClus F → J ∈ Top

Proof

Step Hyp Ref Expression
1 eqid ⊢ ⋃ J = ⋃ J
2 1 isfcls ⊢ A ∈ J fClus F ↔ J ∈ Top ∧ F ∈ Fil ⁡ ⋃ J ∧ ∀ s ∈ F A ∈ cls ⁡ J ⁡ s
3 2 simp1bi ⊢ A ∈ J fClus F → J ∈ Top