big league 
noun
1A group of teams in a professional sport, especially baseball, competing for a championship at the highest level.This marks the first time Percival has changed teams in the big leagues....
- Jack McKeon of the Marlins is another current major league manager who never played in the big leagues.
- Even when he was in college, Bret thought he was better than second basemen in the big leagues.
1.1 (the big league) informal A very successful or important group: the film brought him into the movie world’s big league...
- Aspirants to the big league may note that attractive golf gear costs at least Rs 15,000.
- Two successive hits and they have made it to the big league of scriptwriters in the Malayalam film industry.
- My chance to really hit the big league and gain mainstream popularity.
Derivatives
big leaguer
- I thought before that all the big leaguers were here in Washington.
- Former bush leaguer (when he was with Suicidal Tendencies) Rob Trujilo quits his current gig with big leaguer Ozzy and switches teams when he joins heavy hitters Metallica.
- ‘I got most of the names of Jewish big leaguers from the American Jewish Historical Society,’ he said.