Restriction of Well-Ordering is Well-Ordering
Jump to navigation
Jump to search
Theorem
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 $\preceq'$ is a well-ordering of $T$.
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 is Well-Founded, $\preceq'$ is a well-founded ordering.
Thus $\preceq'$ is a well-ordering.
$\blacksquare$