Condition for Relation to be Transitive and Antitransitive

Theorem
Let $S$ be a set.

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

Then:
 * $\RR$ is both transitive and antitransitive


 * $\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z}$
 * $\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z}$

Necessary Condition
Suppose $\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z}$.

Then $\RR$ is both transitive and antitransitive vacuously.

Sufficient Condition
Suppose $\RR$ is both transitive and antitransitive.

it is not the case that $\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z}$.

Then $\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z$.

By transitivity:
 * $x \mathrel \RR z$

By antitransitivity:
 * $\neg \paren {x \mathrel \RR z}$

This is a contradiction.

Hence we must have $\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z}$.