# Order Isomorphism is Equivalence Relation/Proof 1

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## Theorem

Order isomorphism between ordered sets is an equivalence relation.

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

## Proof

Follows directly from Identity Mapping is Order Isomorphism.

Follows directly from Inverse of Order Isomorphism is Order Isomorphism.

Follows directly from Composite of Order Isomorphisms is Order Isomorphism.

$\blacksquare$

## Sources

- 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 3.2$: Order-preserving mappings. Isomorphisms