Tukey's Lemma

Theorem
Let $\mathcal F$ be a non-empty set of finite character.

Then $\mathcal F$ has an element which is maximal with respect to the set inclusion relation.

Proof
Let $\mathcal N \subseteq \mathcal F$ be a nest.

We will show that $\bigcup \mathcal N \in \mathcal F$.

Let $F$ be a finite subset of $\bigcup \mathcal N$.

By the definitions of subset and of union, each element of $F$ is an element of at least one element of $\mathcal N$.

By the Principle of Finite Choice, there is a mapping $c:F\to\mathcal N$ such that for each $x \in F$, $x \in c(x)$.

Then $f(F)$ is a finite subset of $\mathcal N$.

Since every total ordering of a finite set is a well-ordering, $f(F)$ has a greatest element $P \in \mathcal N \subseteq \mathcal F$.

Then $F$ is a finite subset of $P$, and $P \in \mathcal F$.

Since $\mathcal F$ has finite character, $F \in \mathcal F$.

We have thus shown that every finite subset of $\bigcup \mathcal N$ is in $\mathcal F$. Since $\mathcal F$ is of finite character, $\bigcup \mathcal N \in \mathcal F$.

Thus by Zorn's Lemma, $\mathcal F$ has a maximal element.

Also known as
Tukey's Lemma is also known as the Tukey-Teichmüller Lemma.