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 $A$.

$\blacksquare$

Also see

• Maximal Element of Nest is Greatest Element

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles