Restriction of Well-Ordering is Well-Ordering

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Theorem

Let $S$ be a set or class.

Let $\preceq$ be a well-ordering of $S$.

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

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


Then $\preceq'$ well-orders $T$.


Proof

By the definition of well-ordering, $\preceq$ is a well-founded total ordering.

By Restriction of Total Ordering is Total Ordering, $\preceq'$ is a total ordering.

By Restriction of Well-Founded Ordering, $\preceq'$ is well-founded.

Thus $\preceq'$ is a well-ordering.

$\blacksquare$