# Proof of “AD + every set of reals is Suslin” implies AD$_\mathbb{R}$

Could someone point me toward a proof that "ZF + AD + every set of reals is Suslin" (+ $\mathsf{DC}\_\mathbb{R}$?) implies $\mathsf{AD}\_\mathbb{R}$, either with a reference or a hint?

I am interested how local the proof is; that is, which sets of reals need to be Suslin in order to show the determinacy of a particular real game.

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Sorry about the formatting. I cannot seem to get a subscript \mathbb{R} in the body of the question. –  Trevor Wilson Oct 17 '12 at 16:36
Hi Trevor. Isn't this covered in Richard's paper (More structural consequences of AD)? –  Andres Caicedo Oct 17 '12 at 16:43
Hi Andres, it looks like he just says it is due independently to Martin and Woodin, and is unpublished. –  Trevor Wilson Oct 17 '12 at 16:57
I might be mistaken but if I remember correctly, I saw something like this result in the Cabal reprints (in Steve's Article, volume I). If you don't have them, I'll try to check if it's indeed there later. –  Carlo Von Schnitzel Oct 17 '12 at 20:03
@alephomega It looks like Jackson just mentions that the equivalence between AD$_\mathbb{R}$ and "every set is Suslin" was proved by Woodin. –  Trevor Wilson Oct 17 '12 at 21:37