Theorem bndmet

Description: A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015)

	Ref	Expression
Assertion	bndmet	$$⊢M\in \mathrm{Bnd}\left(X\right)\to M\in \mathrm{Met}\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		isbnd	$$⊢M\in \mathrm{Bnd}\left(X\right)\leftrightarrow \left(M\in \mathrm{Met}\left(X\right)\wedge \forall x\in X\exists y\in {ℝ}^{+}X=x\mathrm{ball}\left(M\right)y\right)$$
2	1	simplbi	$$⊢M\in \mathrm{Bnd}\left(X\right)\to M\in \mathrm{Met}\left(X\right)$$