Identity is only Idempotent Cancellable Element

Theorem
Let $e_S$ is the identity of an algebraic structure $\struct {S, \circ}$.

Then $e_S$ is the only cancellable element of $\struct {S, \circ}$ that is idempotent.

Proof
By Identity Element is Idempotent, $e_S$ is idempotent.

Let $x$ be a cancellable idempotent element of $\struct {S, \circ}$.

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

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