UPDATED 06/08/16 2:09 a.m.

NEW YORK (CBSNewYork/AP) — Democrat Hillary Clinton on Tuesday announced the “milestone” of being the first woman to win a major party’s presidential nomination – even though her opponent, Bernie Sanders, had not conceded.

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Republican Donald Trump likewise pivoted toward the general election, saying a Clinton presidency would continue the “disaster” of President Barack Obama’s years.

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Clinton and Trump both won in New Jersey on Tuesday night.

Clinton also won in New Mexico and South Dakota, while Trump also was swiftly declared the winner in all the other states that held GOP contests Tuesday — South Dakota, New Mexico, Montana, and California.

CBS News estimated early Wednesday morning that Clinton was also likely to win in California, but no final result had been determined as of early Wednesday. Sanders was ahead in Montana, but that race also had not yet been called early Wednesday morning.

In addressing supporters at the Brooklyn Navy Yard, Clinton presented herself as the presumptive nominee and noted that she had now won the majority of pledged delegates.

“It may be hard to see tonight, but we are standing under a glass ceiling right now. But don’t worry – we’re not smashing this one. Thanks to you, we’ve reached a milestone – the first time in our nation’s history that a woman will be a major party’s nominee,” Clinton said.

Clinton also addressed Sanders in a conciliatory tone.

“I want to congratulate Senator Sanders for the extraordinary campaign he has put on,” she said.

Clinton also slammed Trump, saying he wanted to divide the country.

“Donald Trump is temperamentally unfit to be president, and he’s not just trying to build a wall between America and Mexico. He’s trying to wall off Americans from each other,” Clinton said. “When he says, ‘Let’s make America great again,’ that is code for, ‘Let’s take America backwards.’”

Clinton further said Trump “wants to win by stoking fear and rubbing salt in wounds, and reminding us daily just how great he is. Well, we believe we should lift each other up and not tear each other down.”

One supporter, named Matt, made sure he was there to see the speech.

“A couple of weeks ago, my daughter asked me how many girl presidents there had been, and my answer was none,” he told WCBS 880’s Alex Silverman. “And the look on her face was disappointed and shocked.”

Meanwhile, President Obama released a statement late Tuesday congratulating Clinton on securing the nomination, and thanked Sanders for inspiring millions of Americans. Obama also reported that he would be meeting with Sanders on Thursday at Sanders’ request.

While Clinton had enough delegates and superdelegates to clinch the nomination even before the Tuesday primaries, Sanders remains in the race and said late Tuesday that he was not going anywhere.

He said he would be continuing his campaign on to the last primary in Washington, D.C., and then to the convention.

“We are going to fight hard. We are going to fight hard to win the primary in Washington, D.C. And then we take our fight for social, economic, racial and environmental justice to Philadelphia,” Sanders said.

Sanders conceded that he had a “very, very steep fight” ahead, but would “continue to fight for every vote and every delegate.”

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Sanders had been urging superdelegates to drop their support for Clinton before the gathering in Philadelphia, arguing he is a stronger candidate to take on presumptive Republican nominee Donald Trump.

Speaking to reporters, Sanders said he planned to return to Vermont on Wednesday and “assess where we are” following the California results.

He also dismissed the possibility that President Barack Obama could officially back Clinton as early as Wednesday.

“I think it is premature for any endorsement,” he said. “I think it is up to the people to make that decision and it is clear. That’s not me talking. It is the Democratic National Committee that says do not lump together pledged delegates, real delegates with superdelegates who may change their minds.”

Sanders was preparing to speak in Santa Monica, California at 1:10 a.m. Eastern Time.

Meanwhile, Trump also addressed his supporters Tuesday night and took some jabs at Clinton.

“Tonight, we close one chapter in history, and we begin another,” Trump said in his speech in Briarcliff Manor, Westchester County. “Our campaign received more primary votes than any GOP campaign in history, no matter who they are.”

Trump also reached out to Sanders supporters, saying they had been disenfranchised by a “rigged system of superdelegates” on the Democratic side.

“We welcome you with open arms,” Trump said, adding that the trade deals that Sanders spoke against will be handled “far better than anyone thought possible” in a Trump presidency.

But Trump was hardly as forgiving to Clinton – his likely soon-to-be general election opponent.

“The Clintons have turned the politics of personal enrichment into an art form for themselves. They’ve made hundreds of millions of dollars selling access, selling favors, selling government contracts, and I mean hundreds of millions of dollars,” Trump said. “Secretary Clinton even did all of the work on a totally illegal private server – something that how she gets away with this, folks, nobody understands.”

Trump said Clinton would be a “disaster” given how much the country is struggling.

“Our infrastructure is a disaster. Our schools are failing. Crime is rising. People are scared,” Trump said. “The last thing we need is Hillary Clinton in an extension of the Obama disaster.”

Trump said he would give a “major speech” likely on Monday of next week, where he would talk about all the misdeeds of Bill and Hillary Clinton.

“I wonder if the press will want to attend,” Trump said.

As CBS2’s Tony Aiello reported, the solid speech likely reassured Republicans who worried Trump dug a hole with his comments about the Mexican-American judge handling the Trump University case.

Trump has said Judge Gonzalo Curiel, the U.S. District Court judge who is handling the case, can’t be impartial in a legal case involving the businessman because his parents were born in Mexico and Trump wants to build a wall along the border.

Still, Trump has a ways to go with some in the GOP establishment.

“I’m one of those still hoping over the course of the next six months,” said former New York Gov. George Pataki. “Trump can get his act together, lay out a sound political philosophy, embrace the diversity that is America, but it hasn’t happened yet.”

Technically, it’s still not over on either side. Neither Clinton nor Trump will be their parties’ official nominees until the formalities of the delegate votes at the parties’ national conventions.

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(TM and © Copyright 2016 CBS Radio Inc. and its relevant subsidiaries. CBS RADIO and EYE Logo TM and Copyright 2016 CBS Broadcasting Inc. Used under license. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed. The Associated Press contributed to this report.)