Theorem rnblopn

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

	Ref	Expression
Hypothesis	mopni.1	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
Assertion	rnblopn	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge B\in \mathrm{ran}\mathrm{ball}\left(D\right)\right)\to B\in J$$

Proof

Step	Hyp	Ref	Expression
1		mopni.1	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
2	1	blssopn	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to \mathrm{ran}\mathrm{ball}\left(D\right)\subseteq J$$
3	2	sselda	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge B\in \mathrm{ran}\mathrm{ball}\left(D\right)\right)\to B\in J$$