Suppose that $X$ is a set of ordinals. Then $\bigcup X$ is transitive.
Please help me verify my attempt! Many thanks for you!
My attempt:
Suppose that $\alpha\in\beta\in\bigcup X$. Then $\alpha\in\beta\in\gamma$ for some $\gamma\in X$. It follows that $\alpha\in\gamma$ since $\gamma$ is an ordinal and thus transitive. Thus $\alpha\in\bigcup X$. Hence $\bigcup X$ is transitive.