Asymmetric Relation is Antireflexive

Theorem
Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a relation on $S$.

Let $\RR$ be asymmetric.

Then $\RR$ is also antireflexive.

Proof
Let $\RR$ be asymmetric.

Then, by definition:
 * $\tuple {x, y} \in \RR \implies \tuple {y, x} \notin \RR$

$\tuple {x, x} \in \RR$.

Then:

Thus $\RR$ is antireflexive.