Well-Ordering on Set is Proper Well-Ordering

Theorem

Let $\struct {S, \preccurlyeq}$ be a well-ordered set.

Then $\preccurlyeq$ is a proper well-ordering.


Proof

By definition, a proper well-ordering is a well-ordering on a class such that:

every proper lower section of $S$ is a set.


We have a fortiori that a proper lower section of $S$ is a subclass of $S$.

But here we have that $S$ is a set.

The result follows from Subclass of Set is Set.

$\blacksquare$


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
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