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 $$x = e_S$$.