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