Hausdorff's Maximal Principle

Theorem
Each chain in the poset $\left({\mathcal P, \preceq}\right)$ is included in a maximal chain in $\mathcal P$.

Proof
Let $\preceq$ be a partial order on the set $\mathcal P$ and let $X$ be a chain in $\left({\mathcal P, \preceq}\right)$.

By definition, a maximal chain in $\mathcal P$ that includes $X$ is a chain $Y$ in $\mathcal P$ such that $X \subseteq Y$ and there is no chain $Z$ in $\mathcal P$ with $X \subseteq Z$ and $Y \subsetneq Z$.

Let us define $\mathcal C$ as:
 * $\mathcal C = \lbrace Y \vert Y$ is a chain in $\mathcal P$ and $X \subseteq Y \rbrace$

Then: are one and the same.
 * a maximal chain in $\mathcal P$ that includes $X$ and
 * a maximal element of $\mathcal C$ under the partial order induced on $\mathcal C$ by inclusion

It thus suffices to show that $\mathcal C$ contains such a maximal element.

According to Zorn's Lemma, $\mathcal C$ contains a maximal element if each chain in $\mathcal C$ has an upper bound in $\mathcal C$.

Let $W$ be a chain in $\mathcal C$, and let $Z = \bigcup W$.

Let $a, b \in Z$.

Then $\exists A, B \in W: a \in A, b \in B$.

Since $W$ is a chain in $\mathcal C$, one of $A$ and $B$ includes the other.

Suppose WLOG $A \subseteq B$.

Then $a, b \in B$.

Since $B$ is a chain in $\mathcal P$, either $a \preceq b$ or $b \preceq a$.

Therefore $Z$ is a chain in $\mathcal P$.

By definition of $Z$, we have that $\forall Y \in W: X \subseteq Y \subseteq Z$.

Thus $Z$ is an upper bound for $W$ under inclusion.

Finally, since $X \subseteq Z$, $Z \in \mathcal C$.