# Well-Founded Relation Determines Minimal Elements It has been suggested that this article or section be renamed: Well-Founded Relation is Strongly Well-Founded One may discuss this suggestion on the talk page.

## 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

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

If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.

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

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.