Metamath Proof Explorer

Theorem filfinnfr

Description: No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

		Ref	Expression
	Assertion	filfinnfr	$⊢ (F \in Fil (X) \land S \in F \land S \in Fin) \to ⋂ F \neq \emptyset$

Step	Hyp	Ref	Expression
1		filfbas	$⊢ F \in Fil (X) \to F \in fBas (X)$
2		fbfinnfr	$⊢ (F \in fBas (X) \land S \in F \land S \in Fin) \to ⋂ F \neq \emptyset$
3	1 2	syl3an1	$⊢ (F \in Fil (X) \land S \in F \land S \in Fin) \to ⋂ F \neq \emptyset$