Intersection of Elements of Power Set

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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:

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

$\blacksquare$


Sources