Identity 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 the empty set $\varnothing$ serves as the identity for $\left({\mathcal P \left({S}\right), \cup}\right)$.

Proof
First we note that from Empty Set Element of Power Set, we have that $\varnothing \in \mathcal P \left({S}\right)$.

Then from Union with Empty Set: we have:
 * $\forall A \subseteq S: A \cup \varnothing = A = \varnothing \cup A$

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

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

Thus we see that $\varnothing$ acts as the identity for $\left({\mathcal P \left({S}\right), \cup}\right)$.