I'm wondering if a well-ordered set $B$ which is similar to an ordinal $A$ is necessarily an ordinal ?
I think it may be not as the $\in$ relation in the ordinal number does not necessarily be the counter-party order defined in $B$, so therefore $B$ need not be an ordinal. However, I can't think of an example to this. Moreover, the structure on an ordinal number is so special that maybe every well-ordered set similar to it should also be ordinal.
Thank you for your enlightment (and some examples/instruction of proof if possible)!
No. Any set with a single element is well ordered and order-isomorphic to $1$, but not all single-element sets are ordinals.