Order Isomorphism is Equivalence Relation/Proof 1

Proof
Let $\struct {S_1, \preccurlyeq_1} \cong \struct {S_2, \preccurlyeq_2}$ denote that $\struct {S_1, \preccurlyeq_1}$ is isomorphic to $\struct {S_2, \preccurlyeq_2}$.

Checking in turn each of the criteria for equivalence:

Reflexivity
Thus $\cong$ is seen to be reflexive.

Symmetric
Thus $\cong$ is seen to be symmetric.

Transitive
Thus $\cong$ is seen to be transitive.

$\cong$ has been shown to be reflexive, symmetric and transitive.

Hence the result.