Class under Progressing Mapping such that Elements are Sandwiched is Nest

Theorem
Let $A$ be a class.

Let $g: A \to A$ be a progressing mapping on $A$ such that:


 * $\forall x, y \in A: \map g x \subseteq y \lor y \subseteq x$

Then $A$ is a nest:
 * $\forall x, y \in A: x \subseteq y \lor y \subseteq x$

Proof
By definition of progressing mapping:
 * $\forall x \in A: x \subseteq \map g x$

Thus by Subset Relation is Transitive:
 * $\map g x \subseteq y \implies x \subseteq y$

and it follows that:
 * $\forall x, y \in A: x \subseteq y \lor y \subseteq x$

Hence the result by definition of nest.