Proof That the Order Relation for the Ordinal Numbers is a Partial Order Relation
A partial order relation is reflexive, transitive and antisymmetric.
Let
and
be well ordered sets such that
if and only if
is order isomorphic to a subset of
Ifand
then we write
The relation is reflexive is for every ordinal number
Letrepresent a well ordered set such that
The identity function on
is an order isomorphism hence
To prove the transitive property we must show that for any ordinal numbers
Take three well ordered sets such that
and
Ifis order isomorphic to a subset of
andis order isomorphic to a subset of
thenis order isomorphic to a subset of
Hence
Finally, we prove thatand
Sinceand
order isomorphisms exist such that
with
and
with
The composite function
for all
is an order isomorphism ofto a subset of
Hence
and