Let F be a family of sets having properties $(Z_1)-(Z_3)$ for Zorn’s lemma. Assume that F has no maximal element. Then show that for any $A ∈ F$ there is $x \not\in A$ s.t. $A ∪ {x} ∈ F$.
In the textbook it says: Theorem 3.31 (Zorn’s lemma). Let F be a family of sets with the following properties:
(Z1) ∅∈F;
(Z2) if A∈F and $B\subset A$,then B∈F;
(Z3) if $C \subset F$ is a chain,then $\bigcup$C∈F.
Then F has a maximal element; i.e., there is a set A ∈ F such that there is no B ∈ F with $A \subset B$.
However, it is clearly shown that F has to have a maximal element so isnt the question contradicting itself?