Big League (1877)

“It has been let out, that the Evansville club was not organized for the purpose of going on a round of victorious games, but only to play against the big league clubs, to let our citizens have a sight at the way professional muscle can handle the ball and bat.” The Evansville Daily Courier, June 17, 1877, p4

Previous earliest use (Dickson Baseball Dictionary, 3rd edition, 2009):
1882. “The American Base Ball Association threatens to bark savagely at the big league [National League]. Players know on which side their bread is buttered” (The Cleveland Herald, Aug. 28)