Condition for Relation to be Transitive and Antitransitive

Theorem
Let $S$ be a set.

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

Then:
 * $\mathcal R$ is both transitive and antitransitive


 * $\neg \left({\exists x, y, z \in S: x \mathrel {\mathcal R} y \land y \mathrel {\mathcal R} z}\right)$
 * $\neg \left({\exists x, y, z \in S: x \mathrel {\mathcal R} y \land y \mathrel {\mathcal R} z}\right)$