The question is as below:
Let $ X $ be a set and denote $ X_{0} = X $, $ \forall n \in \mathbb{N} $ : $ X_{n+1} = \bigcup X_n $. Using separation axiom, union axiom (by ZF), and recursion, we create the set: $ Y = \bigcup_{n \in \mathbb{N}} X_n $. Prove that $ Y $ is a transitive set.
I know that it is proved by using induction, but I couldnt figure what I need to use induction on. I thought maybe when I start with n = 0, the claim will be for the union of $X_0, X_1$ and but then I'm confused about the required assumption for at the step.
I would like to have some idea. Thanks
Suppose $A\in Y$. Then $A\in X_n$ for some $n\in\mathbb{N}$. And if $a\in A$ then $a\in\bigcup X_n$. And $\bigcup X_n=X_{n+1}$. Hence $a\in X_{n+1}$. And as $n+1\in\mathbb{N}, a\in Y$. Hence $Y $ is transitive.