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

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

Thus $\exists x \in S: \left({x, x}\right) \notin \mathcal R$

and so $\mathcal R$ is not reflexive.

Now suppose $\Delta_S \subseteq \mathcal R$.

Thus $\mathcal R$ is reflexive.

Comment
Some sources use this as the definition of a reflexive relation.