Set is Transitive iff Subset of Power Set

Theorem
A set $S$ is transitive :
 * $S \subseteq \powerset S$

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

Necessary Condition
Let $S$ be transitive.

Let $s \in S$.

By definition of transitive set:
 * $s \subseteq S$

Then by definition of power set:
 * $s \in \powerset S$

Hence, by definition of subset:
 * $S \subseteq \powerset S$

Sufficient Condition
Let $S \subseteq \powerset S$.

Let $s \in S$.

Then by definition of subset:
 * $s \in \powerset S$

By definition of power set:
 * $s \subseteq S$

As this is true for all $s \in S$, it follows by definition that $S$ is transitive.