Boris Johnson has set out how he believes the Tories can win again in 2020, insisting the party must help the poor and drive social mobility.
Mr Johnson, a key contender to replace David Cameron as Tory leader before the 2020 election, said this required expansive Government intervention in transport, housing and education.
Some of Mr Johnson's rivals for the leadership, Chancellor George Osborne and Home Secretary Theresa May, have been widely seen as edging closer to the leadership at the Mayor's expense since May's General Election win.
In a speech for the Centre for Social Justice, Mr Johnson acknowledged a rapidly growing wage gap between top executives and the average salaries of their workers.
But the London Mayor said this was only tolerable if tax evasion was successfully tackled, wages were driven up, and social mobility is kickstarted after decades of decline.
Mr Johnson said: "I believe one nation Tories need to go further. We need to go beyond trickle down and we need to ensure that in a dynamic market economy there is real cohesion, a real sense of shared interest.
"A sense, as David Cameron has so often and so rightly said, that we are all in it together and in my view people will tolerate a wealth gap on the scale we see in Britain today if and only if we satisfy three conditions."
He continued: "As I look forward to the next five years, there could be no more exciting time to be a one nation Tory.There is a huge mandate to transform this country, and above all a chance to transform our attitudes to ourselves and what we can achieve."
Quoting an ancient proverb, the Uxbridge MP said: "That approach is the way we will win and win again for the people of this country."
Mr Johnson said Mr Osborne had been "completely right" to crack down on tax evaders and corporations who offshore profits.
And he praised the "national living wage" announcement, before adding: "Though I stress that there are still millions in this city who can be and should be paid the London living wage of £9.15 an hour.
"And I think it is outrageous there are executives earning about 450 times the average pay of their employees and yet whose firms are mainlining hundreds of millions of pounds of taxpayer funded subsidies in the form of in-work credits that allow them to get away with low pay."
Mr Johnson said it was the job of Conservatives to "be the door bursters" who help people climb the social ladder.
He said: "What is so worrying is we are seeing social mobility in this country declining, it has declined over the last three decades.
"You see it in the law and in journalism and in Parliament, in just about every profession, you see it in acting for heaven's sake - at the Oscars there was competition between an old Etonian and an old Harrovian like a bad joke. You even see it in sport.
"It is decades since we had a culture of bright kids from poor backgrounds in huge numbers bursting down the doors of the establishment."
Mr Johnson said there were "three great forces" available to put people from poor backgrounds on an "equal footing".
The Mayor said there was an "unmistakeable link" between home ownership and economic prosperity, adding "cheap and efficient" transport was crucial to help people get to jobs.
And he said homes and transport were of "no use" if schools could not give children qualifications and aspirations to take on big jobs.