ThmDex – An index of mathematical definitions, results, and conjectures.
Collection of sets ordered by inclusion contains a maximal element
Formulation 0
Let $P$ be a D1687: Inclusion-ordered collection of sets such that
(i) $\max(P)$ is the D4464: Set of maximal elements in $P$
Then \begin{equation} |\max(P)| \geq 1 \end{equation}
Proofs
Proof 0
Let $P$ be a D1687: Inclusion-ordered collection of sets such that
(i) $\max(P)$ is the D4464: Set of maximal elements in $P$
If $C \subseteq P$ is a D848: Chain in $X$, then result R4142: Union is an upper bound to each set in the union shows that the D77: Set union $\cup C$ is an upper bound for $C$ in $P$. Hence, the claim is a consequence of R447: Zorn's lemma. $\square$