Set is Transitive iff Subset of Power Set

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

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

Necessary Condition
Let $a$ be transitive.

Let $x \in a$.

By definition of transitive set:
 * $x \subseteq a$

Then by definition of power set:
 * $x \in \powerset a$

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

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

Let $x \in a$.

Then by definition of subset:
 * $x \in \powerset a$

By definition of power set:
 * $x \subseteq a$

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