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
From Relation is Transitive and Antitransitive iff Relation Null or Singleton, either:
 * $\mathcal R = \varnothing$

or:
 * $\mathcal R = \left\{{\left({x, y}\right)}\right\}$ for some $x, y \in S$.

Let $\mathcal R = \varnothing$.

From Relation both Symmetric and Asymmetric is Null it follows that $\mathcal R$ is asymmetric.

Otherwise, let $\mathcal R = \left\{{\left({x, y}\right)}\right\}$ for some $x, y \in S$.

As $\mathcal R = \left\{{\left({x, y}\right)}\right\}$ it follows that $\left({x, y}\right) \notin \mathcal R$.

That is, $\mathcal R$ is asymmetric.