◀ ▲ ▶Branches / Settheory / Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals
Since ordinals are downward closed, the elements $\alpha, \beta\in X$ are ordinals themselves if $(X,\in_X)$ is an ordinal. This leads to the following lemma, concerning any ordinals.
Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals
For two ordinals $\alpha$ and $\beta$ the following are equivalent:
 $\alpha\subseteq\beta$
 $\alpha\in\beta$ or $\alpha = \beta.$
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References
Bibliography
 Hoffmann, Dirk W.: "Grenzen der Mathematik  Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011