Proof
(related to Corollary: Strictly, Well-ordered Sets and Transitive Sets)
- Let $(V,\prec)$ be a strictly-ordered set, which is a well-order.
- By definition "$\prec$" is well-founded.
- Moreover, we have already shown that every strict order is extensional, therefore "$\prec$" is extensional.
- Since "$\prec$" is both, well-founded and extensional, we can apply the Mostowski's theorem which ensures the existence of a transitive set $X:=\pi[V],$ where $\pi[V]$ is the Mostowski collapse of the Mostowski function $\pi:V\to X$ defined by $$\pi(x):=\{\pi(y)\mid y\in V\wedge y\prec x\}.$$ Moreover, $\pi$ is injective and fulfills the property $$u\prec v\Longleftrightarrow \pi(u)\in_X\pi(v).$$
- Therefore, $\pi$ is an order embedding between $(V,\prec)$ and $(X,\in_X).$
- Thus, $(X,\in_X)$ is also a strictly-ordered set which is well-ordered with respect to the contained relation $\in_X.$
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References
Bibliography
- Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018
