Well-Ordered Class is not Isomorphic to Initial Segment

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

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


Let $a \in A$.

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

Aiming for a contradiction, suppose $\phi: A \to A_a$ 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.


Also presented as

Some sources present this result in the context of well-ordered sets, and do not take into account the concept of classes.

The result is the same whether the domain of any hypothetical order isomorphism is a set or a propr class.