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.