Intersection of Elements of Power Set

Theorem
Let $S$ be a set.

Let:
 * $\ds \mathbb S = \bigcap_{X \mathop \in \powerset S} X$

where $\powerset S$ is the power set of $S$.

Then $\mathbb S = \O$.

Proof
By Intersection is Subset:
 * $\ds \forall X \in \powerset S: \bigcap_{X \mathop \in \powerset S} X \subseteq X$

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

So:
 * $\ds \bigcap_{X \mathop \in \powerset S} X \subseteq \O$

From Empty Set is Subset of All Sets:
 * $\ds \O \subseteq \bigcap_{X \mathop \in \powerset S} X$

So by definition of set equality:


 * $\ds \bigcap_{X \mathop \in \powerset S} X = \O$