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.

$\blacksquare$