Intersection of Elements of Power Set

Theorem
Let $S$ be a set.

Let:
 * $\displaystyle \mathbb S = \bigcap_{X \mathop \in \mathcal P \left({S}\right)} X$

where $\mathcal P \left({S}\right)$ is the power set of $S$.

Then $\mathbb S = \varnothing$.

Proof
By Intersection is Subset:
 * $\displaystyle \forall X \in P \left({S}\right): \bigcap_{X \mathop \in \mathcal P \left({S}\right)} X \subseteq X$

From Empty Set is Element of Power Set:
 * $\varnothing \in P \left({S}\right)$

So:
 * $\displaystyle \bigcap_{X \mathop \in \mathcal P \left({S}\right)} X \subseteq \varnothing$

From Empty Set is Subset of All Sets:
 * $\displaystyle \varnothing \subseteq \bigcap_{X \mathop \in \mathcal P \left({S}\right)} X$

So by definition of set equality:


 * $\displaystyle \bigcap_{X \mathop \in \mathcal P \left({S}\right)} X = \varnothing$