Ordinal Membership is Trichotomy

Theorem

Let $A$ and $B$ be ordinals.

Then:

$\left({A = B}\right) \lor \left({A \in B}\right) \lor \left({B \in A}\right)$

where $\lor$ denotes logical or.

Proof

By Relation between Two Ordinals, it follows that:

$\left({A = B}\right) \lor \left({A \subset B}\right) \lor \left({B \subset A}\right)$

By Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, the result follows.

$\blacksquare$