Power Set of Transitive Set is Transitive

Theorem
Let $S$ be a transitive set.

Then its power set $\powerset S$ is also a transitive set.

Proof
Let $S$ be transitive.

By Set is Transitive iff Subset of Power Set:
 * $S \subseteq \powerset S$

Then by Power Set of Subset:
 * $\powerset S \subseteq \powerset {\powerset S}$

Thus by Set is Transitive iff Subset of Power Set:
 * $\powerset S$ is a transitive set.