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.