Equivalence of Definitions of Reflexive Relation

Theorem
A relation $$\mathcal{R} \subseteq S \times S$$ is reflexive iff it contains the diagonal relation: $$\Delta_S \subseteq \mathcal{R}$$.

Proof

 * Suppose $$\mathcal{R} \subseteq S \times S$$ is reflexive.

Let $$\xi \in \Delta_S$$. From the definition of the diagonal relation, $$\xi$$ is of the form $$\left({x, x}\right)$$ where $$x \in S$$. Thus:


 * Now suppose $$\Delta_S \subseteq \mathcal{R}$$.

Thus $$\mathcal{R}$$ is reflexive.

Not sure about the first part. Seems a bit handwavey to me.