Ordinal Membership is Trichotomy/Proof 2

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: $\S 7.10$