Relation is Symmetric and Antisymmetric iff Coreflexive

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

Then $\mathcal R \subseteq \Delta_S$ where $\Delta_S$ is the diagonal relation.

Proof
Suppose $\mathcal R \not \subseteq \Delta_S$.

Then $\exists \left({x, y}\right) \in \mathcal R: x \ne y$.

But then as $\mathcal R$ is symmetric, we have $\left({y, x}\right) \in \mathcal R$.

So we have $\left({x, y}\right) \in \mathcal R$ and $\left({y, x}\right) \in \mathcal R$ where $x \ne y$.

Thus $\mathcal R$ is not antisymmetric, contrary to hypothesis.

So our assumption that $\mathcal R \not \subseteq \Delta_S$ is false, and the result follows.