Zorn's Lemma/Warning
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Theorem
Formulation 1
Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set such that every non-empty chain in $S$ has an upper bound in $S$.
Then $S$ has at least one maximal element.
Formulation 2
Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set.
Let $T \subseteq \powerset S$ be the set of subsets of $S$ that are totally ordered by $\preceq$.
Then every element of $T$ is a subset of a maximal element of $T$ under the subset relation.
Warning
The statement of Zorn's Lemma supposes the existence of an upper bound in $S$ for any (non-empty) chain $A$.
It does not guarantee the existence of an upper bound for $A$ in $A$ itself.
The conclusion is that:
- $\forall a \in S: a \le x \implies a = x$