Theorem rnblopn

Description: A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006)

	Ref	Expression
Hypothesis	mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion	rnblopn	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐵 ∈ ran ( ball ‘ 𝐷 ) ) → 𝐵 ∈ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	blssopn	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) ⊆ 𝐽 )
3	2	sselda	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐵 ∈ ran ( ball ‘ 𝐷 ) ) → 𝐵 ∈ 𝐽 )