Zero of Power Set with Intersection

Theorem
Let $S$ be a set and let $\powerset S$ be its power set.

Consider the algebraic structure $\struct {\powerset S, \cap}$, where $\cap$ denotes set intersection.

Then the empty set $\O$ serves as the zero element for $\struct {\powerset S, \cap}$.

Proof
From Empty Set is Element of Power Set:
 * $\O \in \powerset S$

From Intersection with Empty Set:
 * $\forall A \subseteq S: A \cap \O = \O = \O \cap A$

By definition of power set:
 * $A \subseteq S \iff A \in \powerset S$

So:
 * $\forall A \in \powerset S: A \cap \O = \O = \O \cap A$

Thus we see that $\O$ acts as the zero element for $\struct {\powerset S, \cap}$.