Nesthood has Finite Character

Theorem
Let $P$ be the property of sets defined as:
 * $\forall x: \map P x$ denotes that $x$ is a nest.

Then $P$ is of finite character.

That is:
 * if $x$ is a nest, then every finite subset of $x$ is a nest.

Proof
A nest is a class on which $\subseteq$ is a total ordering.

Let $x$ be a nest.

Let $y \subseteq x$.

From Restriction of Total Ordering is Total Ordering it follows that $M$ is also a nest.

This holds in particular if $y$ is a finite set.

Hence the result.