# Set Theory/Zorn's Lemma and the Axiom of Choice/Well-founded

< Set Theory | Zorn's Lemma and the Axiom of ChoiceA binary relation *R* is **well-founded** iff for every set *A*

**Theorem**: A binary relation *R* is well-founded iff for every binary relation *S*

**Proof:** Let *R* be a well founded relation and let *S* be a relation such that

Let

and let

Then

It follows that *A* is empty, and therefore

Conversely, suppose that for every relation *S* we have

Let *A* be a set such that

Let and let . Then

It follows that

and so

and consequently