Well I assume the set of natural numbers exist, by the axiom of infinity. From there you can construct the set of rationals. And you can construct the power set of the set of rationals, by the axiom of the power set. And the set of all Dedekind cuts is a subset of the power set of rationals satisfying those properties listed above, which we can construct by the axiom schema of separation. All of this is Zermelo-Frankel set theory, don't even need the axiom of choice.