Order Isomorphism is Symmetric
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Theorem
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {S_1, \preccurlyeq_1}$ be isomorphic to $\struct {S_2, \preccurlyeq_2}$.
Then $\struct {S_2, \preccurlyeq_2}$ is isomorphic to $\struct {S_1, \preccurlyeq_1}$.
Proof
Let $\phi: S_1 \to S_2$ be an order isomorphism from $\struct {S_1, \preccurlyeq_1}$ to $\struct {S_2, \preccurlyeq_2}$.
From Inverse of Order Isomorphism is Order Isomorphism, $\phi^{-1}: S_2 \to S_1$ is an order isomorphism from $\struct {S_2, \preccurlyeq_2}$ to $\struct {S_1, \preccurlyeq_1}$.
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 $24 \ \text {(b)}$