Subset of Well-Ordered Set is Well-Ordered/Proof 2

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

By definition of well-ordered set, $\struct {S, \preceq}$ is:

a totally ordered set

and:

a well-founded set.

By Subset of Toset is Toset, $\struct {T, \preceq'}$ is a totally ordered set.

By Subset of Well-Founded Relation is Well-Founded, $\preceq'$ is a well-founded relation.

Hence the result.

$\blacksquare$


Sources

  • 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $15 \ \text {(b)}$
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