Proof
(related to Corollary: Strictly, Wellordered Sets and Transitive Sets)
 Let $(V,\prec)$ be a strictlyordered set, which is a wellorder.
 By definition "$\prec$" is wellfounded.
 Moreover, we have already shown that every strict order is extensional, therefore "$\prec$" is extensional.
 Since "$\prec$" is both, wellfounded 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 strictlyordered set which is wellordered with respect to the contained relation $\in_X.$
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Hoffmann, D.: "Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise", Hoffmann, D., 2018