Trichotomy of Ordinals (Cantor)

Proof

Let \(\alpha, \beta\) be fixed ordinals and \(\gamma=\alpha\cap \beta\). By the lemma about equivalence of set inclusion and element inclusion of ordinals, it follows from

1. \(\gamma\subseteq\alpha\) that \(\gamma\in\alpha\) or \(\gamma=\alpha\), and from
2. \(\gamma\subseteq\beta\) that \(\gamma\in\beta\) or \(\gamma=\beta\).

So we have four possible cases:

1. \(\gamma\in\alpha\) and \(\gamma=\beta\). From this case it follows that \(\beta\in\alpha\).
2. \(\gamma\in\beta\) and \(\gamma=\alpha\). From this case it follows that \(\alpha\in\beta\).
3. \(\gamma=\beta\) and \(\gamma=\alpha\). From this case it follows that \(\alpha=\beta\), and by the axiom of foundation, neither \(\alpha\in \beta\) nor \(\beta\in \alpha\) is possible.
4. \(\gamma\in\beta\) and \(\gamma\in\alpha\). However, this case is not possible, since otherwise we would have \(\gamma\in\alpha\cap\beta=\gamma\), contradicting the axiom of foundation.

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