Strictly Well-Founded Relation determines Strictly Minimal Elements

Theorem

Let $A$ be a class.

Let $\RR$ be a strictly well-founded relation on $A$.

Let $B$ be a nonempty class such that $B \subseteq A$.

Then $B$ has a strictly minimal element under $\RR$.

Proof

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First a lemma:

Lemma

Let $A$ be a non-empty class.

Let $\RR$ be a strictly well-founded relation on $A$.

Then $A$ has a strictly minimal element under $\RR$.

$\Box$

Let $\RR' = \paren {B \times B} \cap \RR$.

By Restriction of Strictly Well-Founded Relation is Strictly Well-Founded, $\RR'$ is a strictly well-founded relation.

By the lemma:

$B$ has a strictly minimal element $m$ under $\RR'$.

By Minimal WRT Restriction, $m$ is a strictly minimal element under $\RR$ in $B$.

$\blacksquare$

Also see

• Well-Founded Proper Relational Structure Determines Minimal Elements‎
• Proper Well-Ordering Determines Smallest Elements

weaker results that do not require the Axiom of Foundation.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.21$