Theorem metge0

Description: The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006) (Revised by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Assertion	metge0	$$⊢\left(D\in \mathrm{Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to 0\le ADB$$

Proof

Step	Hyp	Ref	Expression
1		metxmet	$$⊢D\in \mathrm{Met}\left(X\right)\to D\in \mathrm{\infty Met}\left(X\right)$$
2		xmetge0	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to 0\le ADB$$
3	1 2	syl3an1	$$⊢\left(D\in \mathrm{Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to 0\le ADB$$