Theorem metgt0

Description: The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of Gleason p. 223 and its converse. (Contributed by NM, 27-Aug-2006)

	Ref	Expression
Assertion	metgt0	$$⊢\left(D\in \mathrm{Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to \left(A\ne B\leftrightarrow 0<ADB\right)$$

Proof

Step	Hyp	Ref	Expression
1		metxmet	$$⊢D\in \mathrm{Met}\left(X\right)\to D\in \mathrm{\infty Met}\left(X\right)$$
2		xmetgt0	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to \left(A\ne B\leftrightarrow 0<ADB\right)$$
3	1 2	syl3an1	$$⊢\left(D\in \mathrm{Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to \left(A\ne B\leftrightarrow 0<ADB\right)$$