  We study the logical structure of Teichm{\"u}ller-Tukey lemma, a maximality
principle equivalent to the axiom of choice and show that it corresponds to the
generalisation to arbitrary cardinals of update induction, a well-foundedness
principle from constructive mathematics classically equivalent to the axiom of
dependent choice.From there, we state general forms of maximality and
well-foundedness principles equivalent to the axiom of choice, including a
variant of Zorn's lemma. A comparison with the general class of choice and bar
induction principles given by Brede and the first author is initiated.
