Power Set is Complete Lattice/Proof 2

Proof
From Set is Subset of Itself:
 * $S \in \powerset S$

Let $\mathbb S$ be a non-empty subset of $\powerset S$.

From Intersection is Subset:
 * $\bigcap \mathbb S \in \powerset S$

Hence, from Set of Subsets which contains Set and Intersection of Subsets is Complete Lattice:


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

where $\bigcap \mathbb S$ is the infimum of $\mathbb S$.