Story highlights President Donald Trump tweeted a congratulatory note to Perez

Perez tweeted back with a note from him and Keith Ellison

Washington (CNN) Newly elected Democratic National Committee Chairman Tom Perez began his new rivalry with President Donald Trump this weekend, crowing that he will be a tougher political foe than Trump anticipates.

The President tweeted a congratulatory note to Perez, the former labor secretary, for winning the race for chair of the Democratic National Committee on Saturday.

"Congratulations to Thomas Perez, who has just been named Chairman of the DNC," the President wrote. "I could not be happier for him, or for the Republican Party!"

Perez shot back with a note from him and Keith Ellison, the liberal Minnesota congressman who lost the DNC chair race but whom Perez named his vice chair.

Call me Tom. And don't get too happy. @keithellison and I, and Democrats united across the country, will be your worst nightmare. https://t.co/fu7WvLofrD — Tom Perez (@TomPerez) February 25, 2017

"Call me Tom. And don't get too happy. @keithellison and I, and Democrats united across the country, will be your worst nightmare."