Coreflexive Relation Subset of Diagonal Relation

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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.

$\blacksquare$