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 e. ( Fil ` X ) /\ S e. F /\ S e. Fin ) -> |^| F =/= (/) )

Proof

Step	Hyp	Ref	Expression
1		filfbas	|- ( F e. ( Fil ` X ) -> F e. ( fBas ` X ) )
2		fbfinnfr	|- ( ( F e. ( fBas ` X ) /\ S e. F /\ S e. Fin ) -> |^| F =/= (/) )
3	1 2	syl3an1	|- ( ( F e. ( Fil ` X ) /\ S e. F /\ S e. Fin ) -> |^| F =/= (/) )