# 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

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

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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 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

weaker results that do not require the Axiom of Foundation.