Subset of Well-Ordered Set is Well-Ordered/Proof 3

Theorem

Let $\struct {S, \preceq}$ be a well-ordered set.

Let $T \subseteq S$ be a subset of $S$.

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

Then the relational structure $\struct {T, \preceq'}$ is a well-ordered set.

Proof

Let $V$ be a basic universe.

By definition of basic universe, $S$ and $T$ are all elements of $V$.

By the Axiom of Transitivity, $S$ and $T$ are both classes.

Thus $T$ is a subclass of $S$.

We have by hypothesis that $\preceq$ is a well-ordering on $S$.

So from Subclass of Well-Ordered Class is Well-Ordered, $\preceq'$ is a well-ordering on $T$.

Hence the result.

$\blacksquare$