Identity is only Idempotent Cancellable Element

Theorem
If $e_S$ is the identity of a monoid $\left({S, \circ}\right)$, then $e_S$ is the only cancellable element of $\left({S, \circ}\right)$ that is idempotent.

Proof
From Identity of Monoid is Cancellable, the identity is cancellable.

Let $x$ be a cancellable idempotent element of $\left({S, \circ}\right)$.

So $x \circ x = x \circ e_S$.

But because $x$ is also by hypothesis cancellable, it follows that $x = e_S$.