Restriction of Well-Founded Ordering is Well-Founded

Theorem

Let $S$ be a set or class.

Let $T$ be a subset or subclass of $S$.

Let $\preceq$ be a well-founded ordering of $A$.

Let $\preceq'$ be the restriction of $\preceq$ to $T$.

Then $\preceq'$ is a well-founded ordering of $T$.

Proof

By Restriction of Ordering is Ordering, $\preceq'$ is an ordering.

By Restriction of Well-Founded Relation is Well-Founded, $\preceq'$ is a well-founded relation on $T$.

Hence the result by definition of well-founded ordering.

$\blacksquare$