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:
 * $\forall x \in F: x \in c \left({x}\right)$

Then $f \left({F}\right)$ is a finite subset of $\mathcal N$.

From Finite Totally Ordered Set is Well-Ordered, $f \left({F}\right)$ 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
The Tukey-Teichmüller Lemma is also known as:
 * The Teichmüller-Tukey Lemma
 * The Teichmüller-Tukey Theorem
 * The Tukey-Teichmüller Theorem
 * Tukey's Lemma.