Idempotent Magma Element forms Singleton Submagma

Theorem
Let $\left({S, \circ}\right)$ be a magma.

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

Then $\left({\left\{{x}\right\}, \circ}\right)$ is a submagma of $\left({S, \circ}\right)$.

Proof
By Singleton Subset, $x \in S \iff \left\{{x}\right\} \subseteq S$.

By the definiton of idempotence, $x \circ x = x \in \left\{{x}\right\}$.

Thus $\left\{{x}\right\}$ is a subset of $S$ which is closed under $\circ$.

By the definition of submagma, the result follows.