Proving the order-isomorphism lemma for ordinals with transfinite induction

[jgens]

I am trying to prove the following results: If α and β are ordinals, then the orderings (α,∈) and (β,∈) are isomorphic if and only if α = β.

So far, I have only proved that the class Ord is transitive and well-ordered by ∈. I can prove this result with the following lemma: If f:α→β is an order-preserving map, then z ≤ f(z). However, I am having difficulty proving this lemma without something like transfinite induction. Any help?

 

[micromass]

Assume that $z\leq f(z)$ does NOT hold. Then there is a least a such that $f(a)<a$. Take f of both sides.

 

[jgens]

That works perfectly! I am silly for not thinking of something like that. Thanks!