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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 ∈ ( 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 ≠ ∅ )