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.