Nesthood has Finite Character/Proof 1

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

if and only if:

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.

$\blacksquare$