Power Set of Subset

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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

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

$\blacksquare$


Sources