Definition:Order Isomorphism/Well-Orderings/Class Theory

Definition
Let $\struct {A, \preccurlyeq_1}$ and $\struct {B, \preccurlyeq_2}$ be well-ordered classes.

Let $\phi: A \to B$ be a bijection such that $\phi: A \to B$ is order-preserving:
 * $\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$

Then $\phi$ is an order isomorphism.