Asymmetric Relation is Antireflexive

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a relation on $S$.

Let $\mathcal R$ be asymmetric.

Then $\mathcal R$ is also antireflexive.

Proof
Let $\mathcal R$ be asymmetric.

Then, by definition:
 * $\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \notin \mathcal R$

Suppose $\left({x, x}\right) \in \mathcal R$. Then:

Thus $\mathcal R$ is antireflexive.