# criterion for a set to be transitive

###### 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. ∎

Title criterion for a set to be transitive CriterionForASetToBeTransitive 2013-03-22 16:18:23 2013-03-22 16:18:23 Wkbj79 (1863) Wkbj79 (1863) 6 Wkbj79 (1863) Theorem msc 03E20 CumulativeHierarchy