Ordinal Membership is Trichotomy/Proof 1

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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

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

Hence:

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

As Ordinal is Transitive, 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.

$\blacksquare$


Sources