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 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin ) → ∩ 𝐹 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		filfbas	⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) )
2		fbfinnfr	⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin ) → ∩ 𝐹 ≠ ∅ )
3	1 2	syl3an1	⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin ) → ∩ 𝐹 ≠ ∅ )