Order-Preserving Bijection on Wosets is Order Isomorphism

Theorem
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be well-ordered sets.

Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving:
 * $\forall x, y \in S: x \mathop {\preceq_1} y \implies \phi \left({x}\right) \mathop {\preceq_2} \phi \left({y}\right)$

Then:
 * $\forall x, y \in S: \phi \left({x}\right) \mathop {\preceq_2} \phi \left({y}\right) \implies x \mathop {\preceq_1} y$

That is, $\phi: S \to T$ is an order isomorphism.

Proof
A well-ordered set is a totally ordered set by definition.

A bijection is a surjection by definition.

The result follows from Order Isomorphism iff Strictly Increasing Surjection.