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:

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

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers