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The loss means Manchester City, which beat Everton 3-0 earlier in the day, can be crowned champions with a win against Chelsea next Sunday.
City has 85 points — four more than Arsenal — and also has a game in hand.
Arsenal only has two more games to play this season and the most points it can get is 87.
City needs only one more victory from the final three games to retain its title, but won’t even need that if Arsenal loses against Nottingham Forest on Saturday.
Arteta admitted pre-match it was a must-win for his team and defeat came after a promising first half.
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Enciso’s 51st-minute header was followed by late efforts from Undav and Estupinan.
Meanwhile, Brighton’s win keeps alive its slim hopes of a top-four finish after moving up to sixth with games in hand on Liverpool, Manchester United and Newcastle.
Arsenal had chances to take the lead in the first half and Leandro Trossard saw an effort clip the crossbar.
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It only took six minutes of the second half for Brighton to go in front through Enciso’s header.
The game was effectively over when Undav struck in the 86th, lobbing a shot over goalkeeper Aaron Ramsdale.
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Brighton manager Roberto de Zerbi sprinted down the touchline to celebrate and was at it again deep into stoppage-time when Estupinan fired home on the rebound after Ramsdale spilled Undav’s effort.