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:
 * $x$ is a nest


 * every finite subset of $x$ is a nest.
 * every finite subset of $x$ is a nest.

Proof
By definition, a nest $N$ is a class on which $\subseteq$ is a total ordering.

Here we are given that $N$ is a set.

The result follows from Property of being Totally Ordered is of Finite Character.