Looks like all the skeptics can put those divorce rumors to bed! Pink's husband Carey Hart has been snapped in New York City for the first time since Pink announced she was moving her family to the Big Apple.

Check out the pics ... the 50-year-old motocross and off-road truck racer and was snapped lugging a huge backpack around the snow-lined streets of NYC Wednesday, bundled up in a black puffer jacket.

Pink and Carey both have homes in L.A. and don't own property in New York -- yet -- so they're currently living in a rental.

This marks the first photographic evidence of the couple's marriage being intact after Pink said as much herself last week. The singer informed fans they had unfortunately fallen victim to "fake news," bashing the split rumors in an Instagram video.

But, in brighter news ... Pink explained the cross-country move while guest-hosting "The Kelly Clarkson Show" yesterday, saying it was so her 14-year-old daughter Willow "could study theater and experience more Broadway."

Pink and Carey have 2 children -- Willow and 9-year-old son Jameson.

Enjoy living in New York ... hopefully, rumor-free!