Connected Equivalence Relation is Trivial

Theorem
Let $S$ be a set.

Let $\mathcal R$ be a relation on $S$ which is both connected and an equivalence relation.

Then $\mathcal R$ is the trivial relation on $S$.

Proof
By definition of equivalence relation, $\mathcal R$ is an equivalence relation :


 * $\Delta_S \cup \mathcal R^{-1} \cup \mathcal R \circ \mathcal R \subseteq \mathcal R$

From Relation is Connected iff Union with Inverse and Diagonal is Trivial Relation:
 * $\Delta_S \cup \mathcal R^{-1} \cup \mathcal R = S \times S$

Hence the result.