I stumbled upon this exercise:
Prove that, if $X$ is an inductive set, then $Y=\{x \in X\colon x \subset X\}$ is inductive.
I easily proved that $\emptyset \in Y$ but then I could not find a way to show that $\forall y \in Y, y \cup \{ y \} \in Y$.
Is every inductive set transitive? Obviously, if every inductive set were transitive, the exercise would be pretty trivial. Could anyone clarify this for me? I would be really grateful.
Thanks!