Transitive and Antitransitive Relation is Asymmetric

Theorem
Let $S$ be a set.

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

Let $\mathcal R$ be both transitive and antitransitive.

Then $\mathcal R$ is asymmetric.

Proof
Let $\left({x, y}\right) \in \mathcal R$ for some $x, y \in S$.

Then as $\mathcal R$ is antitransitive:
 * $\left({x, x}\right) \notin \mathcal R$

and so as $\mathcal R$ is transitive and $\left({x, x}\right) \notin \mathcal R$:
 * $\left({y, x}\right) \notin \mathcal R$

That is, $\mathcal R$ is asymmetric.