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Metamath Proof Explorer

Theorem filfbas

Description: A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009) (Revised by Mario Carneiro, 28-Jul-2015)

		Ref	Expression
	Assertion	filfbas	$⊢ F \in Fil (X) \to F \in fBas (X)$

Step	Hyp	Ref	Expression
1		isfil	$⊢ F \in Fil (X) \leftrightarrow (F \in fBas (X) \land \forall x \in 𝒫 X (F \cap 𝒫 x \neq \emptyset \to x \in F))$
2	1	simplbi	$⊢ F \in Fil (X) \to F \in fBas (X)$