# Ordinal Membership is Transitive

## Theorem

Let $\On$ denote the class of all ordinals.

Then:

$\forall \alpha, \beta, \gamma \in \On: \paren {\alpha \in \beta} \land \paren {\beta \in \gamma} \implies \alpha \in \gamma$

## Proof

By Strict Ordering of Ordinals is Equivalent to Membership Relation the statement to be proved is equivalent to:

$\forall \alpha, \beta, \gamma \in \On: \paren {\alpha \subsetneqq \beta} \land \paren {\beta \subsetneqq \gamma} \implies \alpha \subsetneqq \gamma$

which follows (indirectly) from Subset Relation is Transitive.

$\blacksquare$