Order Isomorphism is Equivalence Relation

Theorem
Isomorphism between posets is an equivalence.

So any given family of posets can be partitioned into disjoint classes of isomorphic sets.

So, two isomorphic posets can be regarded as identical where it is the structure of the partial ordering that is important rather than the elements themselves.

Two isomorphic posets obviously have the same power, as there is a bijection between them by definition.

Proof

 * Reflexive:

Follows directly from the fact that the identity mapping is a poset isomorphism.


 * Symmetric:

Follows directly from the fact that the inverse of a poset isomorphism is itself a poset isomorphism.


 * Transitive:

Follows directly from the fact that the composite of two poset isomorphisms is itself a poset isomorphism.