What are $\in$-well orderable classes?

Question

$V$ is not $\in$-well ordered but $Ord$ is.

Is $Ord$ the unique proper class of the universe which is $\in$-well ordered?

Answer 1

No, $\rm Ord$ is not the unique class which is well-ordered by $\in$, but it is the unique transitive class which is well-ordered by $\in$.

You can concoct all sort of crazy classes which are well-ordered by $\in$ and are not transitive, for example pick any set $x$ and construct something like the ordinals on top of it. Namely $x=0'$ and $\alpha'+1=\alpha'\cup\{\alpha\}$, unions at limit steps.

This will result in a class of sets well-ordered by $\in$. Taking any subclass of such sets will also result in that. For example the class of all limit ordinals is well-ordered by $\in$ as well.

But one can show that every proper class which is well-ordered by $\in$ must be order isomorphic to the ordinals, because every proper initial segment of that class must be a set.