Empty Set is Bottom of Lattice of Power Set

Theorem
Let $X$ be a set.

Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be the lattice of the power set of $X$.

Then:
 * $\O = \bot$

where $\bot$ denotes the bottom of $L$.

Proof
By Empty Set is Subset of All Sets:
 * $\forall S \in \powerset X: \O \subseteq S$

By Empty Set is Element of Power Set:
 * $\O \in \powerset X$

Thus by definition of the smallest element:
 * $\O = \bot$