## Proof That the Order Relation for the Ordinal Numbers is a Partial Order Relation

A partial order relation is reflexive, transitive and antisymmetric.

Letandbe well ordered sets such thatif and only ifis order isomorphic to a subset ofIfandthen we write

The relation is reflexive is for every ordinal numberLetrepresent a well ordered set such thatThe identity function onis an order isomorphism hence

To prove the transitive property we must show that for any ordinal numbers

Take three well ordered sets such thatand

Ifis order isomorphic to a subset ofandis order isomorphic to a subset ofthenis order isomorphic to a subset of

Hence

Finally, we prove thatand

Sinceandorder isomorphisms exist such that

withandwith

The composite functionfor allis an order isomorphism ofto a subset ofHenceand