= Proof outline of Zorn's lemma
Assume the <axiom of choice> and suppose a <poset> satisfying the hypothesis has no <maximal element>. Fix a <choice function> on its nonempty <subsets>. By <transfinite recursion>, choose at each <ordinal> stage an <upper bound> of the <chain in a partial order> already constructed, then choose a strictly larger member. Both choices are available, including the initial empty stage. This produces distinct members throughout the <Hartogs ordinal> of $P$, contradicting the <Hartogs theorem>. Thus a <maximal element> exists. Conversely, apply <Zorn's lemma> to partial <choice functions> on a family of nonempty <sets>, ordered by extension. A <chain in a partial order> has the compatible <union> as an <upper bound>, and a <maximal element> must be defined on the whole family, since one missing value could be adjoined.
<Zorn's lemma> was introduced in 1935 to shorten <algebra> proofs using <choice> or the <well-ordering theorem>.<Zorn’s Lemma and Some Applications by Keith Conrad>[Zorn’s Lemma and Some Applications, §1, p. 4, paragraph on the 1935 introduction to shorten algebraic proofs]{ref}
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