Theorem fbelss

Description: An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015)

	Ref	Expression
Assertion	fbelss	⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝑋 ∈ 𝐹 ) → 𝑋 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		fbsspw	⊢ ( 𝐹 ∈ ( fBas ‘ 𝐵 ) → 𝐹 ⊆ 𝒫 𝐵 )
2	1	sselda	⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝑋 ∈ 𝐹 ) → 𝑋 ∈ 𝒫 𝐵 )
3	2	elpwid	⊢ ( ( 𝐹 ∈ ( fBas ‘ 𝐵 ) ∧ 𝑋 ∈ 𝐹 ) → 𝑋 ⊆ 𝐵 )