Angus Robertson claims proposals for the Irish border opens the door for a single market Scotland
The Prime Minister told MPs the Government does not want to see a "return to the borders of the past" between the two countries.
The SNP's leader in Westminster appeared to try to use Mrs May's comments to bolster his party's call for Scotland to have different arrangements to the rest of the UK after divorce from Brussels.
But Mrs May said Mr Robertson was "trying to imply something that isn't there".
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Mr Robertson said during prime minister's questions: "The Prime Minister had a very successful international visit in this last week to Ireland and there she spoke publicly about her commitment … not to have a hard border on these islands, that there should continue to be free movement of peoples on these islands and that trade should be protected and enhanced.
"Given that people will be watching this, not just in Britain but also in Ireland, would she take the opportunity now to explain how she will deliver these sensible and important outcomes?"
May has called for a frictionless border between the Republic of Ireland and Ulster after Brexit
Mrs May replied: "These are absolutely the outcomes that we want to see.
The Prime Minister had a very successful international visit in this last week to Ireland
"I was very pleased to meet with the Taoiseach and to discuss with him the joint intent that both his government and mine have to ensure that we don't see a return to the borders of the past in Northern Ireland, and just to say that of course we focus on the land border that is between Northern Ireland and the Republic of Ireland, of course the issue of movements from Ireland does affect other places as well.
"It affects ports in Wales and, of course, Stranraer.
Government Loses Brexit Vote Appeal Tue, January 24, 2017
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Britain's most senior judges ruled that Prime Minister Theresa May does not have the power to trigger the formal process Article 50 for the UK's exit from the European Union without Parliament having a say.
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Issued by the Supreme Court of (top row, from the left) Lord Neuberger, Lord Mance, Lord Kerr, Lord Sumption, (bottom row, from the left) Lady Hale, Lord Clarke, Lord Wilson and Lord Hodge, who agreed with the majority decision that the Government could not trigger Article 50 without Parliamentary approval.
The SNP leader in Westminster used Mrs May's comments to bolster his claims for Brexit arrangements
"It is an important issue for us and we have agreed the work that we are going to do together to deliver what I believe will be as frictionless as possible a border and also one of the objectives that I set out in my plan for our negotiating objectives is to retain the common travel area."
The Scottish Government has set out a paper proposing to keep Scotland in the European single market even if the UK leaves, reflecting the majority Remain vote north of the border.
Mr Robertson said: "The Prime Minister has very helpfully explained that it is perfectly possible for parts of these islands to be in the single market, without hard borders, with free movement of people and at the same time protect and enhance trade with one another.
The SNP have set out a paper proposing Scotland remains in the single Market after Brexit
"This is very, very welcome.
"Will the Prime Minister give a commitment to work with the Irish government and a commitment to work with the Scottish government to deliver all of these things or will we just have to get on with it ourselves?"
Mrs May said Mr Robertson should "listen to the answers that are given" rather than "trying to imply something that isn't there".
She said: "Of course, if he is so worried about having a frictionless border between Scotland and countries in the European Union then he shouldn't want to take Scotland out of the European Union by wanting to see it independent."
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