Power Set is Lattice

Theorem
Let $S$ be a set.

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

Then $\struct {\powerset S, \subseteq}$ is a lattice.

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

Let $X, Y \in \powerset S$.

Then from Union is Smallest Superset:
 * $X \subseteq T, Y \subseteq T \iff X \cup Y \subseteq T$

and from Intersection is Largest Subset:
 * $X \subseteq T, Y \subseteq T \iff T \subseteq X \cap Y$

So $X \cap Y$ is the infimum and $X \cup Y$ is the supremum of $\set {X, Y}$.

Hence by definition $\powerset S$ is a lattice.

Also see

 * Power Set is Complete Lattice