set theory - Adding a random real makes the set of ground model reals meager

The proof is based on the fact that there is a decomposition ${bf R}=Acup B$ of the reals such that $A$, $B$ are (very simple) Borel sets, $A$ is meager, $B$ is of measure zero, and ${bf R}=Acup B$ even holds if after forcing we reinterpret the sets. Nos let $s$ be a random real. If $rin {bf R}$ is an old real, then $snotin r+B$, so $sin r+A$, that is, the meager
$s-A$ contains all old reals.