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

A binary relation R is well-founded iff for every set A

$$A\subseteq R[A]\Rightarrow A=\emptyset $$

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

$$S\circ R\subseteq R\circ S \Rightarrow R\cap S^{-1}=\emptyset$$

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

$$S\circ R\subseteq R\circ S$$

Let

$$X=field(R)$$

and let

$$A=dom(R\cap S^{-1})$$

Then

$$A=dom(R\cap S^{-1}) =dom((S\circ R)\cap I_X) \subseteq dom((R\circ S)\cap I_X) =dom(S\cap R^{-1}) =ran(R\cap S^{-1}) \subseteq R[A] $$

It follows that A is empty, and therefore $$R\cap S^{-1}=\emptyset$$

Conversely, suppose that for every relation S we have

$$S\circ R\subseteq R\circ S \Rightarrow R\cap S^{-1}=\emptyset$$

Let A be a set such that

$$A\subseteq R[A]$$

Let $$B=field(R)$$ and let $$S=BxA$$. Then

$$S\circ R=R^{-1}[B]\times A\subseteq B\times R[A]=R\circ S$$

It follows that

$$R\circ I_A= R\cap (A\times B)=R\cap S^{-1}=\emptyset$$

and so

$$R[A]=\emptyset$$

and consequently $$A=\emptyset$$