Invertible Element containing Identity in Power Structure

Theorem
Let $\struct {S, \circ}$ be a magma.

Let identity element $e \in S$ be an identity element of $\struct {S, \circ}$.

Let $\circ_\PP$ be the operation induced on $\powerset S$, the power set of $S$.

Let $X \subseteq S$ such that:
 * $e \in X$
 * $X$ is invertible for $\circ_PP$.

Then $X = \set e$.

Proof
Let $X \subseteq S$ be invertible for $\circ_PP$ and such that $e \in X$.

From Identity Element for Operation Induced on Power Set, the algebraic structure $\struct {\powerset S, \circ_\PP}$ has an identity element $J = \set e$.

We have: