Theorem   A set $X$ is transitive if and only if its power set $\mathcal{P}(X)$ is transitive.

Proof. First assume $X$ is transitive. Let $A \in B \in \mathcal{P}(X)$ Since $B \in \mathcal{P}(X)$ $B \subseteq X$ Thus, $A \in X$ Since $X$ is transitive, $A \subseteq X$ Hence, $A \in \mathcal{P}(X)$ It follows that $\mathcal{P}(X)$ is transitive.

Conversely, assume $\mathcal{P}(X)$ is transitive. Let $a \in X$ Then $\{a\} \in \mathcal{P}(X)$ Since $\mathcal{P}(X)$ is transitive, $\{a\} \subseteq \mathcal{P}(X)$ Thus, $a \in \mathcal{P}(X)$ Hence, $a \subseteq X$ It follows that $X$ is transitive.