Diagonal Relation is Ordering and Equivalence

Theorem
Let $$\left({S, \Delta_S}\right)$$ be a relational structure where $$\Delta_S$$ is the diagonal relation, defined as:


 * $$\forall x, y \in S: \left({x, y}\right) \in \Delta_S \iff x = y$$

Then $$\Delta_S$$ is the only relation on $$S$$ which is both an equivalence and an ordering.

Proof
From Trivial Ordering and Diagonal Relation Equivalence we know that the diagonal relation possesses these properties.

We now need to show that it is the only relation on $$S$$ which possesses these properties.

Both an equivalence and an ordering are reflexive and transitive.

We also have that:
 * An equivalence relation is symmetric;
 * An ordering is antisymmetric.

The result follows directly from Relation both Symmetric and Antisymmetric.