Coreflexive Relation is Subset of Diagonal Relation

Theorem
A coreflexive relation is a subset of the diagonal relation.

Proof
Let $\mathcal R \subseteq S \times S$ be a coreflexive relation.

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

By definition of coreflexive, it follows that $x = y$, and hence $\left({x, y}\right) = \left({x, x}\right)$.

So by definition of the diagonal relation:
 * $\left({x, y}\right) \in \Delta_S$

Hence the result.