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Proof

(related to Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals)

• Let $\alpha,\beta\in (X,\in_X),$ in which $X$ is an ordinal.
• Since ordinals are downward closed, $\alpha$ and $\beta$ are ordinals.

"$\Leftarrow$".

• Let $\alpha\subseteq \beta.$
• If $\alpha=\beta,$ we are done.
• Assume, therefore that $\alpha\subset \beta.$
• We have $\beta$ is an ordinal, $\alpha\subset \beta$ and $\alpha$ is transitive.
• It follows from the equivalent notions of ordinals that $\alpha\in \beta.$

"$\Rightarrow$".

• Let $\alpha\in\beta$ or $\alpha=\beta.$
• If $\alpha=\beta$ then $\alpha\subseteq\beta,$ and we are done.
• Assume, therefore, that $\alpha\in\beta.$
• Since $\beta$ is an ordinal, $\beta$ is transitive, i.e from $x\in\alpha$ and $\alpha\in\beta$ it follows that $x\in \beta$
• Altogether, it follows that $\alpha \subset\beta.$
  ∎

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References

Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
2. Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018