A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $Δ$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $Δ+ 300$ colours, provided $Δ$ is large enough. We show that this bound can be reduced to $Δ+ 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $Δ+ 2$ colours are enough for $Δ\geq 3$.