Power Set is Complete Lattice

Theorem
Let $S$ be a set.

Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the relational structure defined on $\mathcal P \left({S}\right)$ by the relation $\subseteq$.

Then $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is a complete lattice.

Proof
From Subset Relation on Power Set is Partial Ordering, we have that $\subseteq$ is a partial ordering.

We note in passing that for any set $S$:
 * From Supremum of Power Set, $\mathcal P \left({S}\right)$ has a supremum, that is, $S$ itself
 * From Infimum of Power Set, $\mathcal P \left({S}\right)$ has an infimum, that is, $\varnothing$.

These are also the maximal and minimal elements of $\mathcal P \left({S}\right)$.

Let $\mathbb S$ be a subset of $\mathcal P \left({S}\right)$.

Then:
 * From Union Smallest:
 * $\left({\forall X \in \mathbb S: X \subseteq T}\right) \iff \bigcup \mathbb S \subseteq T$


 * From Intersection Largest:
 * $\left({\forall X \in \mathbb S: T \subseteq X}\right) \iff T \subseteq \bigcap \mathbb S$

So $\bigcap \mathbb S$ is the infimum and $\bigcup \mathbb S$ is the supremum of $\left({\mathbb S, \subseteq}\right)$.

Hence by definition $\mathcal P \left({S}\right)$ is a complete lattice.

Also see

 * Power Set is Lattice