I'm reading up on ordinals, and I've come across various definitions for what it means to be an ordinal. The main book that I'm following is Kunen's Set Theory book. He defines an ordinal as a set that is transitive and is well-ordered by $\in$.
I found a couple of exercises in his book that relates his definition of an ordinal to other definitions that I've seen. I'm trying to figure these out to make more sense of the definitions and concepts in Kunen's book and connect his definition with other definitions I've seen. The exercises both assume the Foundation Axiom and are as follows:
$z$ is an ordinal iff $z$ is a transitive set and $\in$ satisfies trichotomy on $z$. Hint: $z \cap ON$ is an ordinal by a previous lemma. Then, use trichotomy to derive a contradiction from an $\in$-minimal element of $z \backslash ON$, where $ON$ denotes the class of all ordinals.
Showing $(\Rightarrow)$ seemed to follow easily since being an ordinal implies transitivity and well-ordering $\in$ implies that $\in$ is well-founded on $z$, implying trichotomy. I'm having trouble setting up the contradiction in $(\Leftarrow)$ by using the hint.
$z$ is an ordinal iff $z$ is a transitive set and all elements of $z$ are transitive sets.
Again, $(\Rightarrow)$ seemed to follow easily since every element of a transitive set is also transitive. As for $(\Leftarrow)$, I'm stuck.
As always, I would really appreciate any help/hints for these exercises.