Coreflexive Relation is Subset of Diagonal Relation

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

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

Let $\tuple {x, y} \in \RR$.

By definition of coreflexive, it follows that $x = y$, and hence $\tuple {x, y} = \tuple {x, x}$.

So by definition of the diagonal relation:
 * $\tuple {x, y} \in \Delta_S$

Hence the result.