Antiassociative Operation has no Idempotent Elements

Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure and let $ \circ$ be antiassociative on $S$.

Then no element of $S$ is idempotent under $ \circ$, that is:


 * $\forall x \in S: x \circ x \ne x$

Proof
Let $a \in S$ and suppose that $a$ is idempotent under $ \circ$, that is:


 * $a \circ a = a$

Then:


 * $\left({a \circ a}\right) \circ a = a \circ a$

and


 * $a \circ \left({a \circ a}\right) = a \circ a$

This contradicts our assumption that $ \circ$ is antiassociative on $S$.