Wikipedia states that
Every well-ordered set (S,<) is order isomorphic to the set of ordinals less than one specific ordinal number [the order type of (S,<)] under their natural ordering.
I've been trying to find proofs on the Internet of this theorem at an appropriate level of rigor, but am entirely at a loss.
The best motivation I've found for defining von Neumann ordinals as the set of all lesser ordinals is that every well-ordering is order isomorphic to the set of all lesser ordinals under the ordinal comparison operation. Since I'm trying to use this theorem as a way to motivate defining ordinals, all I'd like to use in the proof is a supposition that ordinals are canonical representatives of order types, and not worry about how they're concretely defined.
I'm supposing that I've already proven that ordinal comparison is a well-ordering. How might I go about this proof? (Can I even do it without the von Neumann definition?) Thanks!
Clarification: My goal is not to show that every well-ordering is isomorphic to an ordinal. It is to show that every well-ordering is already isomorphic to a set of canonical representations of all lesser order types under ordinal comparison, so we might as well define ordinals as sets of all lesser ordinals, which is the von Neumann construction.