Well-Founded Relation Determines Minimal Elements

From ProofWiki
Jump to: navigation, search


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.


Let $A$ be a non-empty class.

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

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



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 see any proofs that link to this page, please insert this template at the top.

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 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 $B$.


Also see


weaker results that do not require the Axiom of Foundation.