Description: Dependent Choice. Axiom DC1 of Schechter p. 149. This theorem is
weaker than the Axiom of Choice but is stronger than Countable Choice.
It shows the existence of a sequence whose values can only be shown to
exist (but cannot be constructed explicitly) and also depend on earlier
values in the sequence. Dependent choice is equivalent to the statement
that every (nonempty) pruned tree has a branch. This axiom is redundant
in ZFC; see axdc . But ZF+DC is strictly weaker than ZF+AC, so this
axiom provides for theorems that do not need the full power of AC.
(Contributed by Mario Carneiro, 25-Jan-2013)