Ordinal Membership is Trichotomy/Corollary
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Corollary to Ordinal Membership is Trichotomy
Let $\alpha$ be an ordinal.
Let $x, y \in \alpha$ such that $x \ne y$.
- $x \in y$
- $y \in x$
We have that Element of Ordinal is Ordinal.
The result then follows directly from Ordinal Membership is Trichotomy.
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Corollary $1.15$