Washington (CNN) The two most powerful people in Washington -- Donald Trump and Nancy Pelosi -- met for the first time as President and speaker Friday morning at the White House in a futile attempt to end a government shutdown as it hit the two-week mark.

The meeting occurred in wider talks involving congressional leaders on the second day of divided government in the Trump administration. Things didn't go so well.

That's because Pelosi is now the face of the congressional oversight and investigation from which Trump was spared by two years of GOP rule on Capitol Hill and that now threatens his presidency itself.

It's not just that Pelosi and Trump are political and temperamental opposites. Hanging over their most critical relationship is the possibility that she has the power to open impeachment proceedings should special counsel Robert Mueller deliver a damning verdict on Trump.

As she was sworn in Thursday, the California Democrat preached action on boosting the middle class, combating climate change, shielding children brought to the US by undocumented parents, gender equality and opening up government.

But she also fired several warnings at the President, notably by passing legislation to reopen the government that ignored his demands for border wall funding as the first bill of the new Democratic majority.

"We're not doing a wall. Does anybody have any doubt that we are not doing a wall? So that's that," she said at a news conference on Thursday evening.

And playing on Trump's legal vulnerabilities, Pelosi had declined to rule out impeachment in an interview with NBC's "Today" show and said it was an open question whether a president can be indicted.

Trump laid the groundwork for his face-to-face encounter with his new top political foe by cementing his demands for wall funding in a hastily orchestrated appearance in the White House Briefing Room that signaled he was a long way from backing down on the shutdown.

"I have never had so much support as I have in the last week over my stance for border security, for border control and for, frankly, the wall or the barrier," the President said, flanked by a group of Border Patrol Council members.

Poor omens

When Trump and then-House Minority Leader Pelosi last met, earlier this week at the White House, things headed south when the President ordered his homeland security secretary, Kirstjen Nielsen, to brief lawmakers on border security. At one point, Trump's frustration boiled over at Senate Minority Leader Chuck Schumer and the President lobbed a glowing personal letter from North Korean tyrant Kim Jong Un in the direction of the New York Democrat.

That massive political disconnect is one reason why expectations were so low for Friday's meeting, along with the fact that at this point neither side seems to have an overwhelming incentive to make a deal. After the two-hour session in the White House Situation Room ended, Pelosi described it as "a lengthy and sometimes contentious conversation" but indicated some progress was made.

"How do you define progress in a meeting?" she asked. "When you have a better understanding of each other's position? When you eliminate some possibilities? If that's the judgment, we made some progress."

The speaker has no desire to celebrate her new majority by giving Trump a win on the wall -- the symbol of his 2016 election victory, which broke Democratic hearts. Her position is supported by Schumer, whose minority party votes are needed to pass Senate funding legislation.

Trump triggered the shutdown while facing a revolt from the conservative commentators who help keep his vital political base engaged -- a constituency at which Thursday's "briefing" appeared to be aimed . Climbing down now would expose him to even more of a backlash.

That's why it's tough to see a way out of the showdown that could allow Trump and Pelosi to both claim victory and build goodwill that might be used in pursuit of bipartisan initiatives -- for instance, an infrastructure package, or in future must-do items like budget bills or raising the debt ceiling.

Little in common

JUST WATCHED Trump congratulates House Speaker Pelosi Replay More Videos ... MUST WATCH Trump congratulates House Speaker Pelosi 02:08

Trump and Pelosi could hardly be more different, so it's hard to see how they can forge any kind of bond in the weeks ahead.

Pelosi is a political lifer, steeped in the family business from an early age. Trump owes his stunning electoral success to the fact that he is not a politician at all.

The speaker, by securing a second run at the job despite early rumblings of dissent among younger Democrats, proved once again her mastery of Washington process. Her vote-counting prowess is especially remembered for pushing Obamacare into law during her first turn at running the House.

Trump is a legislative neophyte, and despite enjoying a Republican monopoly on power for two years, his unpredictable nature was often more of a hindrance to his party in advancing its agenda than a help.

Pelosi is no stranger to fighting Republican presidents, even if memories are dimming of her bitter battles with President George W. Bush over funding the Iraq War.

Despite their gaping differences, Pelosi and Bush had a cordial personal relationship. Their mutual respect was crucial to navigating the 2008 financial crisis and was in evidence when they hugged at former President George H.W. Bush's lying in state last month.

Trump sought to build a personal rapport with Pelosi by congratulating her on her election as speaker.

"It's a very, very great achievement, and hopefully we're going to work together, and we're going to get lots of things done," he said.

Yet Trump is a far more volatile personality than Bush and often sees political differences in deeply personal terms. His default mode of projecting power is to identify an enemy and applying withering political fire.

Dan Meyer, who served Bush as deputy assistant for legislative affairs and also experienced shutdowns while working for Republican House Speaker Newt Gingrich in the 1990s, said the political dynamics of new majorities were often shaped by the issues that brought them to power.

"There is a different set of issues that brought this majority to power -- the most significant being the Democrats' resistance to President Trump," Meyer said.

"This is going to be much more difficult to do and not have it be personal. They are going to go after him hard," he said.

If that is how it goes, Pelosi will be ready, said Nadeam Elshami, an ex-chief of staff to Pelosi.

"She has the power to back it up. She has gone toe-to-toe with this President in the past two years," Elshami said on CNN's "New Day," citing his former boss's "resolve, her ability to understand the issues and her ability to look at the President and say, 'Mr President, you are wrong. These are not the facts.' "

Dictating terms

JUST WATCHED Schumer: GOP using shutdown as hostage Replay More Videos ... MUST WATCH Schumer: GOP using shutdown as hostage 02:19

The differing approaches of Pelosi and Trump were on display Thursday.

Pelosi was legislating, passing the bill to reopen the government that put Republicans -- who had already passed a similar effort to keep the government open last month in the Senate -- in a difficult spot.

Trump, meanwhile, was presiding over the propaganda stunt in the White House Briefing Room, designed to change the subject from Pelosi's day of triumph and to provide footage for conservative news channels.

The Pelosi approach is designed to show that Democrats can govern and can provide stability after two years of chaos fostered by Trump's disruption.

The President appears to be betting that the longer the shutdown drags on, the more pressure Democrats will feel to end it as hundreds of thousands of federal workers go unpaid.

Or perhaps a prolonged shutdown could cause voters to sour on both parties and let the President off the hook.

Trump's political team is trying to shape that outcome by portraying Pelosi as of an extreme band of liberals who are soft on immigration and blinded by a ravenous desire to destroy his presidency.

"I think it's unfortunate that you've got an incoming speaker already talking about indictments and impeachment when this President is talking about border security and infrastructure," Trump's counselor Kellyanne Conway said Thursday.

"She obviously came here yesterday not serious about border security, so it doesn't surprise me that she would be so partisan and political on day one. That's just too bad," Conway said.