Ordinal Membership is Trichotomy/Proof 2
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Theorem
Let $\alpha$ and $\beta$ be ordinals.
Then:
- $\paren {\alpha = \beta} \lor \paren {\alpha \in \beta} \lor \paren {\beta \in \alpha}$
where $\lor$ denotes logical or.
Proof
By Relation between Two Ordinals, it follows that:
- $\paren {\alpha = \beta} \lor \paren {\alpha \subset \beta} \lor \paren {\beta \subset \alpha}$
By Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, the result follows.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.10$