The mark of eight no-hitters was set in 1884, the first season overhand pitching was allowed. There haven't been more than seven no-hitters in a season since 1900.
Left-hander Matt Moore shut down the Reds for six innings and was lifted after 76 pitches. Right-hander Hector Neris pitched a 1-2-3 seventh before Bradley gave up Stephenson's homer, cutting the Phillies' lead to 3-1.
Stephenson drove a 3-1 pitch the opposite way to right field, sending it over outfielder Bryce Harper's head and into the seats.
The Chicago Cubs threw the majors' most recent no-hitter with a combined effort June 24. The other no-hitters this year were thrown by San Diego's Joe Musgrove (April 9), Carlos Rodón of the Chicago White Sox (April 14), Baltimore's John Means (May 5), Cincinnati's Wade Miley (May 7), Detroit's Spencer Turnbull (May 18) and the New York Yankees' Corey Kluber (May 19).
In addition, Arizona's Madison Bumgarner pitched a seven-inning hitless game in a doubleheader on April 25 that is not recognized as a no-hitter by Major League Baseball.
Most of those gems were thrown before MLB cracked down on the use of sticky foreign substances by pitchers in late June.
Moore struck out eight and walked two. The six scoreless innings lowered the 32-year-old's ERA to 6.07.
The Reds could have gotten a hit leading off the second, but shortstop Didi Gregorius snared Joey Votto's hard-hit grounder by diving to his right and then firing to second for the force out. Moore coasted from that point with routine outs or strikeouts until being relieved after batting in the sixth.
The Phillies have 13 no-hitters in their history. The last was thrown by Cole Hamels on July 25, 2015.
Bryce Harper ejected in Phillies' loss
Bryce Harper is sent his marching orders in the Phillies' blowout loss to the Reds.