Proof That The Order Relation is Reflexive and Transitive on the Ordinal Numbers

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