----- > [!proposition] Proposition. ([[well-ordering of products]]) > If $A$ and $B$ are [[well-ordered set|well-ordered sets]], then $A \times B$ well-ordered in the [[dictionary order relation|dictionary order]]. > [!proof]- Proof. ([[well-ordering of products]]) > Let nonempty $X \subset A \times B$. Let $Y \subset A$ consist of all *first coordinates* of elements in $X$. $A$ is [[well-ordered set|well-ordered]], so $Y$ has a smallest element $a_{0}$. Then the collection $\{ b : (a_{0}, b) \in X\}$ has a smallest element $b_{0}$, since $B$ is well-ordered. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```