Ordinal Membership is Trichotomy/Proof 1

Proof
From Class of All Ordinals is Well-Ordered by Subset Relation, $\On$ is a nest.

Hence:
 * $\forall \alpha, \beta \in \On: \paren {\alpha \subsetneqq \beta} \lor \paren {\beta \subsetneqq \alpha} \lor \paren {\alpha = \beta}$

From Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, this is equivalent to:


 * $\forall \alpha, \beta \in \On: \paren {\alpha \in \beta} \lor \paren {\beta \in \alpha} \lor \paren {\alpha = \beta}$

Hence the result.