Properties of an L-cardinal (Q1069934)

\(\kappa\) is an L-cardinal if for every \(\lambda >\kappa\), and every formula \(\phi\) there is an elementary embedding \(j: V\to M\) such that: 1) \(j(\kappa)>\lambda\), 2) \(\kappa\) is the first ordinal moved, 3) for every x in \(R(j(\kappa))^ M\), \(M\vDash \phi (x)\to V\vDash \phi (x)\). Some properties of an L-cardinal are mentioned, e.g. \(R_{\kappa}\prec V\). A guess: such a cardinal does not exist?