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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.

criterion for a set to be transitive is owned by Warren Buck.