Antiassociative Operation has no Idempotent Elements

Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

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

Aiming for a contradiction, suppose $a \in S$ such that $a$ is idempotent under $\circ$.

That is:

$a \circ a = a$

Then:

$\paren {a \circ a} \circ a = a \circ a$

and

$a \circ \paren {a \circ a} = a \circ a$

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

$\blacksquare$