Restriction of Strict Well-Ordering is Strict Well-Ordering

Theorem
Let $R$ be a strict well-ordering of $A$.

Let $B \subseteq A$.

Then $R$ is a strict well-ordering of $B$.

Proof
By Restriction of Foundational Relation is Foundational, $R$ is a foundational relation on $B$.

By Totally Ordered Subset, $R$ is a total ordering on $B$.

By the above two statements, $R$ is a strict well-ordering of $B$. {[explain|Indicate the specific definition which identifies that a foundation relation which is a total ordering is a well-ordering.}}

Also see

 * Restriction of Foundational Relation is Foundational