= Zorn's lemma
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{wiki}
= Zorn lemma
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{synonym}
Let $(P,\leq)$ be a nonempty <partially ordered set>. If every <chain in a partial order> $C\subseteq P$ has an <upper bound> in $P$, then $P$ has a <maximal element>. The hypothesis means
$$
\forall C\subseteq P\text{ that is a chain},\quad
\exists u\in P\quad\forall c\in C,\ c\leq u.
$$
The <upper bound> may depend on $C$ and need not belong to $C$. A <least upper bound> is not required. The conclusion asserts that some $m\in P$ satisfies $m\leq x\Longrightarrow x=m$; it does not assert a <greatest element>, which would satisfy $x\leq m$ for every $x\in P$.<How to use Zorn’s lemma by Timothy Gowers>[How to use Zorn’s lemma, “General discussion”: statement and distinction between maximal and bigger than everything else]{ref}
Here a <chain in a partial order> may be the <empty set>. Every member of $P$ is an <upper bound> of the empty <chain in a partial order>, so including it in the hypothesis already forces $P$ to be nonempty. If “chain” means only nonempty <chains in a partial order>, the separate nonempty-$P$ assumption is essential: the empty <poset> otherwise satisfies the hypothesis vacuously and has no <maximal element>. These are two equivalent conventions for the theorem.
The strength of <Zorn's lemma> is that compatible partial constructions can have a completed, unextendible member without an explicit procedure for finding it. The next examples separate the <upper bound> calculation from the reason a <maximal element> has the desired property. All <chains in a partial order> must be covered, including uncountable ones, rather than just increasing <sequences>. The condition is sufficient, not necessary: a <poset> can have a <maximal element> and still have an unbounded <chain in a partial order> elsewhere.
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