Maximal Element under Subset Relation need not be Greatest Element

Theorem
Let $A$ be a class.

Let $M \in A$ be a maximal element of $A$ under the subset relation

Then $M$ is not necessarily the greatest element of $A$.

Proof
Let $A = \set {x, y}$ such that:
 * $x = \set \O$
 * $y = \set {\set \O}$

Then:
 * $x$ and $y$ are both maximal elements of $A$ by definition.

However:
 * $x \not \subseteq y$

and:
 * $y \not \subseteq x$

and so neither $x$ nor $y$ are the greatest element of $A$.

Also see

 * Maximal Element of Nest is Greatest Element