The Axiom of Determinacy implies Dependent Choices in mice

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We show that the Axiom of Dependent Choices, $\operatorname{DC}$, holds in countably iterable, passive premice construced over their reals which satisfy the Axiom of Determinacy, $\operatorname{AD}$, in a $\operatorname{ZF}+\operatorname{DC}_{\mathbb{R}}$ background universe. This generalizes an argument of Kechris for $L(\mathbb{R})$ using Steel’s analysis of scales in mice. In particular, we show that for any $n \leq \omega$ and any countable set of reals $A$ so that $M_n(A) \cap \mathbb{R} = A$ and $M_n(A) \vDash \operatorname{AD}$, we have that $M_n(A) \vDash \operatorname{DC}$. Furthermore, we argue that for countable premice it suffices to work in a background universe satisfying $\operatorname{ZF} + \operatorname{AC}_{\omega,\mathbb{R}}$.