Order Isomorphism is Reflexive

Theorem

Let $\struct {S, \preccurlyeq}$ be an ordered set.

Then $\struct {S, \preccurlyeq}$ is isomorphic to itself.

Proof

Let $I_S: S \to S$ denote the identity mapping on $S$.

From Identity Mapping is Order Isomorphism, $I_S: \struct {S, \preccurlyeq} \to \struct {S, \preccurlyeq}$ is an order isomorphism.

The result follows.

$\blacksquare$

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $24 \ \text {(a)}$