Relation Isomorphism preserves Ordering

Theorem
Let $\struct {A, \RR}$ and $\struct {B, \SS}$ be relational structures which are relationally isomorphic.

Let $\struct {A, \RR}$ be an ordered set.

Then $\struct {B, \SS}$ is also an ordered set.

Proof
Let $\struct {A, \RR}$ be an ordered set.

Recall the definition:

From Relation Isomorphism Preserves Reflexivity:
 * $\SS$ is reflexive.

From Relation Isomorphism Preserves Antisymmetry:
 * $\SS$ is antisymmetric.

From Relation Isomorphism Preserves Transitivity:
 * $\SS$ is transitivie.

So by definition $\struct {B, \SS}$ is an ordered set.