Order-Preserving Bijection on Wosets is Order Isomorphism

Theorem

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be well-ordered sets.

Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving:

$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$

Then:

$\forall x, y \in S: \map \phi x \preceq_2 \map \phi y \implies x \preceq_1 y$

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.

$\blacksquare$