Equivalence of Definitions of Coreflexive Relation

Definition 1 implies Definition 2
Let $\RR$ be a relation which fulfils the condition:
 * $\forall x, y \in S: \tuple {x, y} \in \RR \implies x = y$

Then:

Hence $\RR$ is coreflexive by definition 2.

Definition 2 implies Definition 1
Let $\RR$ be a relation which fulfils the condition:
 * $\RR \subseteq \Delta_S$

Then:

Hence $\RR$ is coreflexive by definition 1.