Her victory in a confidence vote of Conservative MPs late Wednesday made her immune from further party leadership challenge for a year, but she was forced to admit she would quit by 2022.
May will remain as prime minister after winning more than half of the vote, with 117 voting against her tonight.
Her decision to defer the vote sparked a new wave of letters of no confidence which pushed the total beyond the threshold of 48 needed to trigger a ballot.
"That's a very febrile electorate and possibly a duplicitous one".
"The prime minister, through the mandate she secured from the parliamentary party last night, now has the time to have those discussions with European colleagues", he said, adding that the direction of travel was "positive".
Britain's weakened Prime Minister Theresa May has headed for Brussels to lobby European leaders for help after she survived a parliamentary mutiny that highlighted the deadlock over Brexit. And a good thing too, her supporters will quietly think to themselves, because defeat in that vote is certain.
She said that ousting her and holding a leadership vote - a process that could take weeks - could result in Brexit being delayed or even halted.
DUP leader Arlene Foster, who met May shortly before the vote, insisted "tinkering around the edges" of the PM's deal would not be enough to win her party's support for the PM's Brexit deal.
The source played down suggestions the PM, who has been Conservative leader since July 2016, would set a specific date for her departure.
He went on: "Penny Mordaunt voted for Brexit, which in the Conservative Party is helpful.
She can not get her deal through, her very bad and unsatisfactory withdrawal agreement its not supported by the DUP by many Conservatives or by the Opposition".
Lib Dem leader Sir Vince Cable, who is against Brexit, told BBC Breakfast: "We are still back with the problem that the government has a proposal that we can't get through Parliament and we have got to try and break that gridlock".
"Nearly 40 percent of those who know Theresa May best, her own Conservative MPs, have voted to say they have no confidence in her as prime minister". The EU will likely give her some rhetorical cover for facilitating implementation of the withdrawal agreement-but they have ruled out any changes to the agreement itself.
Corbyn's number two - shadow chancellor John McDonnell - hinted that Labour could call the motion next week. "But of course, the political pressure doesn't go away for her".
"I don't expect an immediate breakthrough but what I do hope is that we can start to work as quickly as possible on the assurances that are necessary", she said.
Questions are also being asked about May's promise not to lead the Conservatives into an election in 2022.
The DUP has branded the backstop "unacceptable" and has repeatedly said it would not accept any additional Northern Ireland-only customs checks, because it is concerned that any differences between NI and Great Britain could threaten the union.
That is probably because she has already agreed to it, and so has the EU27.
May survived a confidence vote of her own party's lawmakers late on Wednesday, but she admitted as she arrived at the European Union summit that she will not fight the next general election planned for 2022. "But it would also be very hard to find anyone to replace her". "She is in office, but not in power".
But Liberal Democrat leader Vince Cable said, despite the "high drama" of Wednesday, "nothing has really changed".
A general election will be triggered in the next 12 months as Brexit will end Theresa May's premiership, Number 10′s former policy chief has predicted.
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