Well-Ordering on Class is not necessarily Proper

Theorem

Let $A$ be a class.

Let $\preccurlyeq$ be a well-ordering on $A$.

Then it is not necessarily the case that $\preccurlyeq$ is a proper well-ordering.

Proof

Proof by Counterexample:

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries: Exercise $1.2$