Identity 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 $S$ serves as the identity for $\struct {\powerset S, \cap}$.

Proof

We note that by Set is Subset of Itself, $S \subseteq S$ and so $S \in \powerset S$ from the definition of the power set.

From Intersection with Subset is Subset‎, we have:

$A \subseteq S \iff A \cap S = A = S \cap A$

By definition of power set:

$A \subseteq S \iff A \in \powerset S$

So:

$\forall A \in \powerset S: A \cap S = A = S \cap A$

Thus we see that $S$ acts as the identity element.

$\blacksquare$