Antitransitive Relation is Antireflexive

Theorem
Let $S$ be a set.

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

Let $\RR$ be antitransitive.

Then $\RR$ is also antireflexive.

Proof
Suppose $\RR \subseteq S \times S$ is not antireflexive.

Then $\exists x \in S: \tuple {x, x} \in \RR$. (In the case of $\RR$ being reflexive, the property holds for all $x \in S$.)

Thus $\RR$ is not antitransitive:

... which means $\exists x \in S$ such that the condition for antitransitivity is broken.

So $\RR \subseteq S \times S$ has to be antireflexive.