Zero of Power Set with Union

Theorem
Let $S$ be a set and let $\mathcal P \left({S}\right)$ be its power set.

Consider the algebraic structure $\left({\mathcal P \left({S}\right), \cup}\right)$, where $\cup$ denotes set union.

Then $S$ serves as the zero element for $\left({\mathcal P \left({S}\right), \cup}\right)$.

Proof
We note that from Subset of Itself, $S \subseteq S$ and so $S \in \mathcal P \left({S}\right)$ from the definition of the power set.

From Union with Superset is Superset‎, we have:
 * $A \subseteq S \iff A \cup S = S = S \cup A$.

By definition of power set:
 * $A \subseteq S \iff A \in \mathcal P \left({S}\right)$

So:
 * $\forall A \in \mathcal P \left({S}\right): A \cup S = S = S \cup A$

Thus we see that $S$ acts as the zero.