The AdS instability conjecture, suggested by Dafermos and Holzegel in 2006, states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes. Following the work of Bizon and Rostworowski in 2011, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing mainly on the simpler setting of the spherically symmetric Einstein--scalar field system.      In this talk, we will provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein--massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein--null dust system, allowing for both ingoing and outgoing dust. In order to overcome the "trivial" break down occuring once the null dust reaches the centre $r=0$, we will study the evolution of the system in the exterior of an inner mirror with positive radius $r_{0}$ and prove the conjecture in this setting. After presenting our proof, we will briefly explain how the main ideas can be extended to more general matter fields.