Proof
(related to Proposition: Ordinals Are Downward Closed)
 Let $X$ be an ordinal.
 From the equivalent notions of ordinals it follows that if $w\in X$ then $w$ is a transitive set and a proper subset $w\subset X.$
 Since all elements of $x\in X$ are transive, this holds also for the elements $x\in X\cap w.$
 Therefore, all elements $x\in w$ are transitive.
 This means that $w$ is an ordinal.
∎
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References
Bibliography
 Hoffmann, Dirk W.: "Grenzen der Mathematik  Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011