Oh my days! Just quit it mate, you're utterly mixing things up yet again!
Did you even follow football well enough from this said decade ?
Robinho joined City in 2008, spent less than 2 seasons and played 41 games in total, Carlos Tevez joined City in 2009/10 and Man City finished 5th in the league, their best league finish inside the top 10 after a long while.
Negredo didn't join Man City until 2013 and played only 32 games for Man City in EPL (one season era ?)
Between 2005/06 - 2009/10, Man City best finish in the PL was 5th position, other years the finished 9th and outside the top 10, except your saying Man City became a solid part of the supposed big four between 2011 and 2013, because SAF retired from Man United at the end of the 2012-13 and Man City had only won the league once then, a season prior, and that was the only season they finished above Man United in the said decade before Sir Alex Fergusson retirement. Note; Pellegrini joined Man City after Sir Alex had left Man United.
I don't know where you get your information from but if you're relying on your photographic memory, then you're mixing a lot of things up and might need to cross check in on number of things before you say them.
Man City have only won the league twice between the said decade, one was 2011/2012 which they won the final day of the league finishing with equal points as Man United, only one of the league was during Sir Alex tenure at Man United, and it was Robert Mancini as City manager, and that was the only league Man City won during Sir Alex tenure at Man United, the other league was won by Pellegrini in 2013/14, and that was post Ferguson era.