City have NEVER won five league Manchester derbies in a row. In terms of four in a rows, that's now only ever happened twice - the last four, and on a run between March 1969 and December 1970.
As an extra note - Manuel Pellegrini is now the first City manager to have won his first three Manchester derbies. In fact, he's the first manager to do that for either side - as no United manager has even won their first three derbies.