# Power Set of Subset

## Theorem

Let $S \subseteq T$ where $S$ and $T$ are both sets.

Then:

$\powerset S \subseteq \powerset T$

where $\powerset S$ denotes the power set of $S$.

## Proof

 $\ds X$ $\in$ $\ds \powerset S$ $\ds \leadsto \ \$ $\ds X$ $\subseteq$ $\ds S$ Definition of Power Set $\ds \leadsto \ \$ $\ds X$ $\subseteq$ $\ds T$ as $S \subseteq T$: Subset Relation is Transitive $\ds X$ $\in$ $\ds \powerset T$ Definition of Power Set

$\blacksquare$