Another day at the Emirates Stadium; another mutiny to reflect the bitter frustration among the Arsenal fanbase. This time the target was not Arsène Wenger, the beleaguered manager, but Stan Kroenke, the majority shareholder. “Stan Kroenke, get out of our club” was the cry on the hour and it would get a series of airings, each one louder than the previous.

This was a seismic moment in Arsenal’s modern history. Nobody really expected Liverpool or Manchester City to slip up and allow Wenger and his team the chance to nip in and secure qualification to the Champions League, and they did not. And so, for the first time since 1997-98, Wenger and his club must move into a season without the spur of participation in Europe’s elite competition.

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Arsenal did what they had to do on the field – even after Laurent Koscielny was sent off in the 14th minute for a reckless challenge on Enner Valencia. The captain will be suspended for the FA Cup final against Chelsea on Saturday while Gabriel Paulista will surely be missing, too. The centre-half was taken off on a stretcher after jarring his left knee. “It doesn’t look too good,” Wenger said.

Arsenal led through Héctor Bellerin and Alexis Sánchez and although Everton pulled one back through Romelu Lukaku’s penalty and infuriated their manager, Ronald Koeman, with moments of profligacy – a theme of their season – Arsenal deserved the victory. Aaron Ramsey sealed it with a beautiful curler into the far, top corner in stoppage-time.

The Kroenke Out sentiment was the takeaway from another emotional day and it followed the publicising of his decision to reject a takeover bid worth £1bn from Alisher Usmanov, who holds a 30% stake in the club. If the timing of that story on Friday night was designed to push Kroenke under an uncomfortable spotlight, it achieved its ends.

Arsenal still have the Cup final and they have won eight of their past nine games, including the semi-final against Manchester City. Yet there could be no mistaking the flatness of the mood as the players ambled around their post-match lap of appreciation. Wenger watched from the mouth of the tunnel and this has to go down as his worst and most trying league campaign.

Wenger had promised that his team would fight until the end and they did so with 10 men, after Koscielny had gone through Valencia with a side-on slide tackle. It did not matter that there looked to be no malice. The challenge was late and there is no leeway when a defender gets it wrong like this.

Arsenal had scored early through Bellerin’s first goal of the season and it served to spare the blushes of his team‑mate Danny Welbeck, who had contrived to miss his kick from two yards out, following Mesut Özil’s cross from the left, and with an open goal gaping. Bellerin managed to shoot, on the rebound, into the still empty net.

Koeman made a change on 26 minutes, sending on Ross Barkley and switching to a 4-4-2 formation. He knew his decision to start Barkley on the bench would set the tongues wagging, in the context of the midfielder’s contractual stand-off, and he said afterwards that he did not know if he would re-sign. Barkley is about to enter the final year on his deal. “I will speak to the board tomorrow and we will make a statement about that,” Koeman said.

Moments after Barkley’s introduction, Arsenal extended their lead and it was a bizarrely simple goal, after Sánchez’s shot had hit Phil Jagielka and broken to Welbeck. Jagielka stayed down, Sánchez moved towards the six-yard box, nobody tracked him, Welbeck squared and Sánchez side-footed home. Sánchez would work Joel Robles with a free-kick and, while the Liverpool game remained goalless Arsenal could hope.

The closing stages of the first half were defined by Everton pressure and there was one moment of pinball inside the Arsenal area that finished with Petr Cech clutching gratefully in front of Lukaku. The Arsenal goalkeeper was further extended by Idrissa Gueye and Lukaku while the latter dragged a shot wide. The hammer blow for Arsenal, though, was the news at the end of the half that Liverpool had gone in front.

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Robles made a double save early in the second half from Welbeck and Özil while Gabriel suffered his misfortune when he stretched into a tackle on Valencia only to get his left leg stuck under his own body.

To add the insult, Michael Oliver would book him for the lunge and there were moments when the referee struggled to keep control. Oliver was correct, though, to award Everton their penalty, after Nacho Monreal had leaned towards Leighton Baines’s cross with his arm. Lukaku guided his kick past Cech.

By then, Liverpool were 3-0 up and the mood inside the Emirates Stadium had darkened, with the anti-Kroenke chants underscoring the occasion. Lukaku had a goal ruled out for climbing on the substitute Per Mertesacker, while Özil and another substitute, Alex Iwobi – on two occasions – could not finish. Ramsey showed them how but, by then, the focus had turned to off-the-field issues.