Scouts had been whispering throughout spring training that he could no longer catch up with a big-league fastball.
Despite that, science rarely becomes part of big-league politicking.
Given the paucity of good big-league players, the opportunity usually arrives more likely sooner than later.
Slang definitions & phrases for big-league
Serious; important; professional: No more fooling around, now it's big-league stuff(1940s+)
The Dictionary of American Slang, Fourth Edition by Barbara Ann Kipfer, PhD. and Robert L. Chapman, Ph.D. Copyright (C) 2007 by HarperCollins Publishers. Cite This Source
Idioms and Phrases with big-league
An area of tough competition and high rewards; the largest or foremost of its kind. For example, Winning an Oscar put this unknown actress in the big league. The term alludes to the major (big) leagues of American baseball.
[ Late 1800s