Set whose every Finite Subset is Nest is also Nest

Theorem
Let $x$ be a set of sets with the property that:
 * every finite subset of $x$ is a nest.

Then $x$ is a nest.

Proof
Let $x$ be a set of sets with the given property.

Let $a, b \in x$ be arbitrary.

Then:
 * $\set {a, b} \subseteq x$

and so $\set {a, b}$ is a nest.

That is:
 * $a \subseteq b$ or $b \subseteq a$

As $a$ and $b$ are arbitrary, it follows that:
 * $\forall a, b \in x: a \subseteq b$ or $b \subseteq a$

The result follows from Subset Relation is Ordering.