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

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

$A\subseteq R[A]\Rightarrow A=\mathrm{\varnothing }$

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

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

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 $R\cap S^{-1}=\mathrm{\varnothing }$

Conversely, suppose that for every relation S we have

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

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

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

and so

$R[A]=\mathrm{\varnothing }$

and consequently $A=\mathrm{\varnothing }$