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?