Well-Ordered Class is not Isomorphic to Initial Segment

Theorem
Let $\struct {A, \preccurlyeq}$ be a well-ordered class.

There exists no order isomorphism from $\struct {A, \preccurlyeq}$ to an initial segment of $A$.

Proof
Let $a \in A$.

Let $A_a$ be the initial segment of $A$ determined by $a$.

$\phi: A \to A_a$ is an order isomorphism from $A$ to $A_a$.

From Order Automorphism on Well-Ordered Class is Forward Moving:
 * $a \preccurlyeq \map \phi a$

But by definition of initial segment:
 * $\map \phi a \notin A_a$

Hence $\phi$ cannot be an order isomorphism.

Hence by Proof by Contradiction there exists no such order isomorphism.