**Proof**: *(related to "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 extentional, 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 $\in_X.$

q.e.d

| | | | created: 2019-03-07 16:17:30 | modified: 2019-03-07 16:33:50 | by: *bookofproofs* | references: [8055]

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[8055] **Hoffmann, D.**: “Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise”, Hoffmann, D., 2018

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