Antitransitive Relation is Antireflexive

Theorem
Every relation which is antitransitive is also antireflexive.

Proof
Suppose $\mathcal R \subseteq S \times S$ is not antireflexive.

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

Thus $\mathcal R$ is not antitransitive:

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

So $\mathcal R \subseteq S \times S$ has to be antireflexive.