◀ ▲ ▶Branches / Set-theory / 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 $\alpha = \beta.$

Table of Contents

Proofs: 1

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Proofs: 1

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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