Symmetric and Antisymmetric Relation is Transitive

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a relation in $S$ which is both symmetric and antisymmetric.

Then $\mathcal R$ is transitive.

Proof
Let $\left({x, y}\right), \left({y, z}\right) \in \mathcal R$.

By Relation is Symmetric and Antisymmetric iff Subset of Diagonal Relation:
 * $x = y, y = z$

and so trivially:
 * $\left({x, z}\right) = \left({x, x}\right) \in \mathcal R$

Thus $\mathcal R$ is transitive.