Non-Empty Set of Type M has Maximal Element

Theorem

Let $S$ be a non-empty set of sets which is of type $M$.

Then $S$ has a maximal element under the subset relation.

Proof

Let $S$ be a non-empty set type $M$ set.

Let $x \in S$ be arbitrary.

Then by definition $x$ is a subset of a maximal element of $S$ under the subset relation.

Hence there has to actually be such a maximal element of $S$ for $x$ to be a subset of.

$\blacksquare$

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