Theresa May has insisted she is not sad that Britain is leaving the EU, as she made a direct appeal to the public to back her deal, saying: “The British people don’t want to spend any time arguing about Brexit.”

The prime minister was speaking after EU leaders met for a special summit in Brussels and formally endorsed her Brexit deal, following a discussion lasting less than an hour.

Jean-Claude Juncker, the president of the European commission, said Sunday was a “sad day”.

But asked if she shared the sentiment, May said, “No: but I recognise that others do; I recognise that some European leaders are sad at this moment.”

Play Video 0:35 Theresa May says she does not feel 'sad' about UK leaving the EU - video

Despite the increasingly bleak prospects for her deal passing the House of Commons next month, in what she called “one of the most significant votes for many years”, May struck a defiant note.

She said she had always rejected the “counsel of despair” from those who believed a beneficial Brexit agreement could not be struck.

Despite the fact it is MPs, not voters, who hold her fate in their hands in the coming weeks, she said she wanted to “speak directly to the British people”, and would be embarking on a campaign in the next few days, to sell the deal to the public.

The outlines of that campaign were clear as she described the merits of the deal on Sunday, highlighting an end to free movement “once and for all”, halting “vast annual payments” to the EU, and ending the jurisdiction of the European court of justice as the advantages of her deal.

She also stressed the government’s keenness to move on to domestic priorities – a view she believes is shared by many voters. “The British people don’t want to spend any more time arguing about Brexit,” she said.

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May pointedly highlighted the £394m a week her government has promised to pour into the NHS. That pledge, which the prime minister has previously claimed is the result of a “Brexit dividend”, echoes the £350m a week the Vote Leave campaign claimed could be diverted to the the health service if Britain left the EU.

In an open letter on Sunday, May promised to put her “heart and soul” into winning over MPs and voters to the merits of her deal, which she previously said she believed “with every fibre of my being” was the right one.

At Sunday’s press conference, she again refused to rule out resigning if the deal was rejected by parliament next month, saying: “It’s not about me.”

The foreign secretary, Jeremy Hunt, conceded on Sunday that the odds of the deal passing were “looking challenging” and refused to rule out the risk that the government could collapse if parliament rejected it.

He told BBC One’s The Andrew Marr Show: “It’s not possible to rule out anything, and that’s why all of us have to do is say, what do your constituents actually want in this situation, and we have to work out what’s in the national interest, and it’s all about the balance of risks. This isn’t a perfect deal for everyone, but does have a lot of what everyone wants.”

May echoed the insistence by several EU leaders on Sunday that there was no possibility of reopening negotiations if parliament said no.