Point not in Subset of Metric Space iff Distance from Elements is Greater than Zero

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $H \subseteq A$ be an arbitrary subset of $A$.

Let $x \in A$ be arbitrary.

Then:
 * $x \notin H$


 * $\forall y \in H: \map d {x, y} > 0$
 * $\forall y \in H: \map d {x, y} > 0$