Well-Founded Relation Determines Minimal Elements

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Theorem

Let $A$ be a class.

Let $\prec$ be a foundational relation on $A$.

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


Then $B$ has a $\prec$-minimal element.


Lemma

Let $A$ be a non-empty class.

Let $\prec$ be a foundational relation on $A$.


Then $A$ has a $\prec$-minimal element.


Proof

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Let $\prec' = \left({B \times B}\right) \cap {\prec}$.

by Restriction of Foundational Relation is Foundational, $\prec'$ is a foundational relation

By the lemma:

$B$ has a $\prec'$-minimal element $m$.

By Minimal WRT Restriction, $m$ is $\prec$-minimal in $B$.

$\blacksquare$


Also see

and

weaker results that do not require the Axiom of Foundation.


Sources