Big Leagues (1884)

“The numerous clubs in the country are training schools for players, and talented ones ought to improve their chances to get into the big leagues, which will be the only survivors some day.” The Chittenden Reporter [Jericho, VT], August 13, 1884, p6

Previous earliest use (Dickson Baseball Dictionary, 3rd edition, 2009):
1889. “The [Lee Base Ball Club] is one of the foremost of amateur clubs, and during its existence has turned out many fine ball tossers, who are now members of the big leagues” (Brooklyn Eagle, Dec. 12; Peter Morris).