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 Restriction of Total Ordering is Total Ordering, $R$ is a total ordering on $B$.

By the above two statements, $R$ is a strict well-ordering of $B$.

Also see

 * Restriction of Foundational Relation is Foundational