If both the union and the power set of a set are transitive, does the set have to transitive?

If $x$ is a set, and $\mathcal{P}x$ and $\cup x$ are both transitive sets, does $x$ necessarily have to be transitive?

No. To find a counterexample, consider $x'$ which is transitive, and take $x=x'\setminus\{\varnothing\}$.
• Say, $x=\{1\}$. Then $\mathcal P(x)=\{0,x\}$. – Andrés E. Caicedo Mar 1 '14 at 17:33