Power Set is Complete Lattice

Theorem
Let $S$ be a set.

Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on the power set $\powerset S$ of $S$ by the relation $\subseteq$.

Then:
 * $\struct {\powerset S, \subseteq}$ is a complete lattice

where for every subset $\mathbb S$ of $\powerset S$:
 * the infimum of $\mathbb S$ necessarily admitted by $\mathbb S$ is $\ds \bigcap \mathbb S$.

Also see

 * Power Set is Lattice