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 Strictly Well-Founded Relation is Strictly Well-Founded, $R$ is a strictly well-founded 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 $B$.

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$\blacksquare$

Also see

• Restriction of Strictly Well-Founded Relation is Strictly Well-Founded