Prikry-Silver forcing $\mathbb{V}$ (sometimes just Silver forcing) is the forcing notion consisting of all partial functions $p:\omega\rightarrow 2$ with co-infinite domain. In "Combinatorics on ideals and forcing with trees" Marcia Groszek mentions (without proof) that a Prikry-Silver real has minimal real degree, but not minimal degree.

That the Prikry-Silver real $r$ has minimal real degree means that whenever $s$ is a real in $V[r]$ that doesn't belong to $V$ we have $V[r]=V[s]$. A proof of this can be extracted from some more general results in the seminally named "Combinatorics on ideals and forcing" by Serge Grigorieff.

That $r$ doesn't have minimal degree means that there is some object $A\in V[r]$ for which $V[A]$ is different from both $V$ and $V[r]$. Can anyone point me to a proof of this fact?