I learnt how to construct a well ordered class of oridinals, ORD which satisfies 2 conditions. 1. If A is any well ordered set, there exists $a$$\in$ORD such that A is similar to $a$. 2. If A is a well ordered set and $a$,$b$ $\in$ORD, then if A is similar to $a$ and A is similar to $b$ then $a$=$b$.
The book im studying constructed a class satisfying those 2 conditions, which is $\in$-well-ordered and each element of the class is also $\in$-well-ordered.
My question is that can any well ordered class satisfying those 2 conditions be $\in$-well-ordered?
Since the book keeps emphasizing that 'Which' class of ordinal numbers it is doesnt matter at all. (i.e. 1meter can be measured by a ruler but can also be measured by 1meter-pencil)
I want to know this because I can prove that 'union of a family of ordinals is equal to supremum of the family of the ordinals' by using the property of the $\in$-well-ordered class of ordinals (class of ords, the one I constructed), but I do not know whether that holds for another class of ordinal numbers which satisfies above 2 conditions..