Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation

Theorem
Let $S$ be a set.

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

Then:
 * $\mathcal R$ is reflexive, symmetric and antisymmetric


 * $\mathcal R$ is the diagonal relation $\Delta_S$.
 * $\mathcal R$ is the diagonal relation $\Delta_S$.

Necessary Condition
Let $\mathcal R$ is reflexive, symmetric and antisymmetric.

By definition of reflexive:
 * $\Delta_S \subseteq \mathcal R$

From Relation is Symmetric and Antisymmetric iff Coreflexive:
 * $\mathcal R \subseteq \Delta_S$

By definition of set equality:
 * $\mathcal R = \Delta_S$

Sufficient Condition
Let $\mathcal R = \Delta_S$.

From Relation is Reflexive and Coreflexive iff Diagonal:
 * $\mathcal R$ is reflexive

and
 * $\mathcal R$ is coreflexive.

From Relation is Symmetric and Antisymmetric iff Coreflexive it follows that $\mathcal R$ is both symmetric and antisymmetric.

Hence the result.