Distance Function for Distinct Elements in Metric Space is Strictly Positive

Theorem
Let $A$ be a set.

Let $d: A \times A \to \R$ be a real-valued function on $A$ with the following properties:

which can be considered as being an alternative formulation of the metric space axioms.

Then:
 * $\forall x, y \in A: x \ne y \implies d \left({x, y}\right) > 0$

which is metric space axiom $(M4)$.

Thus $d$ is a distance function, so making $M := \left({A, d}\right)$ a metric space.