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 $A=\emptyset$

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