In the days after the European referendum, while political leaders were still in shock, some of them suggested that Britain might need a cross-party approach to leaving the EU. Peter Mandelson and Yvette Cooper said that the Conservatives would not be able to manage Brexit on their own.

So it has turned out, although it has taken two years and a botched election to prove it. The all-day cabinet meeting at Chequers last week was historic. Theresa May’s Brexit model merely set out the implications of what had already been agreed – in particular, the “backstop” terms for the Irish border – but it was the point at which people had to decide. It was one of those subterranean shifts when British politics realigned.

David Davis and Boris Johnson’s resignations were only the outward show of this change. As Daniel Finkelstein wrote in The Times, they resigned because they had “simply given up”.

They had no alternative to the prime minister’s plan, so they quit because they realised, eight months later, that they could not accept the open Irish border which they had agreed in December.

The fury of the Brexit irreconcilables was finally unleashed. Theresa May was braced for the onslaught, and she knew – because she can count – that she could survive it. She had also taken the precaution, I am told, of sending a message to European capitals asking them not to shoot down the Chequers deal straight away. They didn’t, which is a hopeful sign.

But the ferocity of the cries of betrayal was still shocking, and tempted some Tory irreconcilables – who cannot count – to think they could get rid of her.

Help was at hand in the unlikely form of a graceless interview with the president of the US, who told The Sun she had done Brexit all wrong: “I actually told Theresa May how to do it but she didn’t agree, she didn’t listen to me.”

Even Trump realised he had gone too far and apologised, a procedure that caused him such dissonance that his news conference with May the next day was a Jekyll-and-Hyde performance.

Dr Jekyll read out a prepared text saying the US still hoped to do a UK trade deal, but when he mentioned Brexit, Mr Hyde interjected: “I don’t know what they’re going to do, but whatever you do is okay with me … we will see how that goes.”

Trump did May a huge favour. His discourtesy helped discredit his friend Johnson and helped galvanise those who want to hug the EU close after Brexit. Given that three-quarters of the British people have a negative view of Trump, his attack on May’s handling of Brexit helped the realignment by which a large chunk of Remainers realise that the Chequers model is their best hope.

This applies to Tory and Labour MPs alike. Remember that most Tory MPs voted Remain, and that most Labour MPs don’t share Jeremy Corbyn’s historical Euroscepticism.

Some Labour MPs who paid close attention to the Chequers statement and Thursday’s white paper point out that it meets the six tests set by Keir Starmer, the shadow Brexit secretary. Or, given that one of them is to obtain the “exact same benefits” of EU membership, it comes as close as possible to meeting them.

So while Corbyn might never vote for a Brexit that keeps EU rules banning state aid to industry, most of his MPs might come to a different view when facing the moment of decision.

This is what must be borne in mind whenever it is observed that there is no majority for any kind of Brexit in the House of Commons. There may not be but, at each stage, when MPs are presented with a binary choice, there has to be a majority for one or the other.

The final vote will come down to a choice between a deal and no deal. And when people say there is no majority in the Commons for a no-deal Brexit, what they mean is that there is a majority for almost any deal, even if it means accepting free movement between the UK and EU.

If David Davis were serious about constructing a decision tree – his favourite aid to logical thinking, according to his friend Paul Goodman of Conservative Home – he would see that the Chequers model is the best of the range of realistic outcomes.

Brexit threatens life on the Irish border: in pictures Show all 15 1 /15 Brexit threatens life on the Irish border: in pictures Brexit threatens life on the Irish border: in pictures An abandoned shop is seen in Mullan, Co Monaghan. The building was home to four families who left during the Troubles. The town was largely abandoned after the hard border was put in place during the conflict. Mullan has seen some regeneration in recent years, but faces an uncertain future with Brexit on the horizon Reuters Brexit threatens life on the Irish border: in pictures A defaced ‘Welcome to Northern Ireland’ sign stands on the border in Middletown, Co Armagh Reuters Brexit threatens life on the Irish border: in pictures Mervyn Johnson owns a garage in the border town of Pettigo, which straddles the counties of Donegal and Fermanagh. ‘I’ve been here since 1956, it was a bit of a problem for a few years. My premises has been blown up about six or seven times, we just kept building and starting again,’ Johnson said laughing. ‘We just got used to it [the hard border] really but now that it’s gone, we wouldn't like it back again’ Reuters Brexit threatens life on the Irish border: in pictures Farmer Gordon Crockett’s Coshquin farm straddles both Derry/Londonderry in the North and Donegal in the Republic. ‘At the minute there is no real problem, you can cross the border as free as you want. We could cross it six or eight times a day,’ said Crockett. ‘If there was any sort of obstruction it would slow down our work every day’ Reuters Brexit threatens life on the Irish border: in pictures John Murphy flies the European flag outside his home near the border village of Forkhill, Co Armagh Reuters Brexit threatens life on the Irish border: in pictures Potter Brenda McGinn stands outside her Mullan, Co Monaghan, studio – the former Jas Boylan shoe factory which was the main employer in the area until it shut down due to the Troubles. ‘When I came back, this would have been somewhere you would have driven through and have been quite sad. It was a decrepit looking village,’ said McGinn, whose Busy Bee Ceramics is one of a handful of enterprises restoring life to the community. ‘Now this is a revitalised, old hidden village’ Reuters Brexit threatens life on the Irish border: in pictures Union Flag colours painted on kerbstones and bus-stops along the border village of Newbuildings, Co Derry/Londonderry Reuters Brexit threatens life on the Irish border: in pictures Grass reflected in Lattone Lough, which is split by the border between Cavan and Fermanagh, seen from near Ballinacor, Northern Ireland Reuters Brexit threatens life on the Irish border: in pictures Donegalman David McClintock sits in the Border Cafe in the village of Muff, which straddles Donegal and Derry/Londonderry Reuters Brexit threatens life on the Irish border: in pictures An old Irish phone box stands alongside a bus stop in the border town of Glaslough, Co Monaghan Reuters Brexit threatens life on the Irish border: in pictures Billboards are viewed from inside a disused customs hut in Carrickcarnon, Co Down, on the border with Co Louth in the Republic Reuters Brexit threatens life on the Irish border: in pictures Seamus McQuaid takes packages that locals on the Irish side of the border have delivered to his business, McQuaid Auto-Parts, to save money on postal fees, near the Co Fermanagh village of Newtownbutler. ‘I live in the south but the business is in the North,’ said McQaid. "I wholesale into the Republic of Ireland so if there’s duty, I’ll have to set up a company 200 yards up the road to sell to my customers. I’ll have to bring the same product in through Dublin instead of Belfast’ Reuters Brexit threatens life on the Irish border: in pictures A disused Great Northern Railway line and station that was for customs and excise on the border town of Glenfarne, Co Leitrim Reuters Brexit threatens life on the Irish border: in pictures Alice Mullen, from Monaghan in the Republic of Ireland, does her shopping at a former customs post on the border in Middletown, Co Armagh. ‘I’d be very worried if it was a hard border, I remember when people were divided. I would be very afraid of the threat to the peace process, it was a dreadful time to live through. Even to go to mass on a Sunday, you’d have to go through checkpoints. It is terribly stressful,’ said Mullen. ‘All those barricades and boundaries were pulled down. I see it as a huge big exercise of trust and I do believe everyone breathed a sigh of relief’ Reuters Brexit threatens life on the Irish border: in pictures A bus stop and red post box stand in the border town of Jonesborough, Co Armagh Reuters

It is bound to be modified in negotiations with the EU before October, but even then most Tory MPs and most Labour MPs ought to prefer it to the rigours of leaving the EU without a deal.

Peter Mandelson and Yvette Cooper were right to call for cross-party cooperation two years ago. It is going to happen, whether the leaders of either party acknowledge it or not.

With the loss of her irreconcilables, Theresa May needs Labour votes to get whatever Brexit deal she negotiates through the Commons. Who knows how many Tory MPs would rather leave without a deal – 80, 100. It could be anything up to 158, or half her parliamentary party. Easily enough to wipe out her tiny majority. But there will always be enough Labour MPs willing to put what they see as the national interest first.