Empty Set is Bottom of Lattice of Power Set

Theorem
Let $X$ be a set.

Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be the lattice of power set of $X$.

Then $\varnothing = \bot$

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

Proof
By Empty Set is Subset of All Sets:
 * $\forall S \in \mathcal P\left({X}\right): \varnothing \subseteq S$

By Empty Set is Element of Power Set:
 * $\varnothing \in \mathcal P\left({X}\right)$

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