Labour could be prepared to vote against Theresa May’s final Brexit deal, even if it means walking through voting lobbies with Conservative hard Brexiters, with party sources insisting the government cannot force a no-deal situation.

The party has said it will vote against the final Brexit deal unless it meets shadow Brexit secretary Keir Starmer’s six tests on protection of jobs and rights – in practice, highly unlikely.

“We will vote down the deal unless it meets our six tests,” a party source said. “Even if we are walking down the corridor with Tory Brexiters, we are still voting on the lines we have set ourselves.”

The source said Labour did not believe defeating any final Brexit agreement would set a course for the UK to leave with no deal, which they said was made clear by the “meaningful vote” amendment to the EU withdrawal bill.

“She cannot treat a defeated deal as an endorsement of no deal. Two-thirds of the Tory party would oppose it,” the source said.

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Speaking in the Commons earlier, the Labour leader, Jeremy Corbyn, tore into the prime minister’s strategy, saying jobs and investment were now “a sub-plot in the Tory party civil war” after the resignations of David Davis and Boris Johnson.

Writing for the Guardian, Starmer said there was an “irreconcilable divide within the Conservative party” and the impasse could no longer be resolved as an internal Conservative party debate.

“The agreement struck at Chequers last Friday is flawed in many respects, but at least it appeared to be the first realisation by the prime minister that the approach she adopted two years ago was wrong,” he said. “Her red lines began to yield just a bit. But even this modest shift has proved too much for some in the cabinet and the Conservative party.”

In the Commons, May was quick to raise Labour’s own internal divisions with Corbyn. “He talks about resignations, can I just remind him, I think he’s had 103 resignations from his frontbench, so I’ll take no lectures from him,” she said.

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Some Brexit-backing Labour MPs, including Frank Field, welcomed the Chequers’ deal in the Commons on Monday, saying May should now “seek support from elected leaders in Europe, rather than bureaucrats in Brussels”.

Downing Street has calculated it will need Labour MPs if the deal has a chance of passing in November. It began laying the groundwork on Monday when opposition MPs were invited to a briefing on the Chequers proposal, scheduled to be presented by the Downing Street chief of staff, Gavin Barwell. Starmer encouraged Labour MPs to attend to make their criticism of the deal known.

Barwell never made it to the meeting, one of the first signs that Boris Johnson’s resignation as foreign secretary was imminent. Instead, the presentation was led by the Cabinet Office minister, David Lidington, and Jonathan Black, one of the directors at the cabinet office’s Europe unit. The committee room in the Houses of Parliament was crammed with peers and MPs but, just minutes into the briefing, it descended into chaos when word of Johnson’s resignation emerged.

It was the most surreal meeting ever. All anyone is talking about is Boris. It’s like The Thick of It.

MPs leaving the room said they were convinced that Lidington had not known that Johnson had resigned while the briefing was ongoing. “There was a moment during the presentation when it flashed on everyone’s phones, everyone burst out laughing. And then someone showed him the message,” one said. Lidington was tight-lipped as he left the room.

“What a joke, what a joke,” one senior Labour MP muttered on the way out.

Another described it as “the most surreal meeting ever” as they left the room. “They are presenting a deal which is collapsing in front of our eyes. The projector doesn’t work. All anyone is talking about is Boris. It’s like The Thick of It.”