Well-Ordered Classes are Isomorphic at Most Uniquely

Theorem
Let $\struct {A, \preccurlyeq_A}$ and $\struct {B, \preccurlyeq_B}$ be well-ordered classes.

Then there exists at most one order isomorphism from $\struct {A, \preccurlyeq_A}$ to $\struct {B, \preccurlyeq_B}$

Proof
Let $\phi$ and $\psi$ be order isomorphisms from $A$ to $B$.

Then from Inverse of Order Isomorphism is Order Isomorphism:
 * the inverse mapping $\psi^{-1}$ is an order isomorphism from $B$ to $A$.

Hence from Composite of Order Isomorphisms is Order Isomorphism:
 * $\phi \circ \psi^{-1}$ is an order isomorphism from $A$ to $A$.

But from Order Automorphism on Well-Ordered Class is Identity Mapping:
 * $\phi \circ \psi^{-1}$ is the identity mapping.

From Composite of Mapping with Inverse of Another is Identity implies Mappings are Equal:
 * $\phi = \psi$