Looks like Campsis radicans/Trumpet Creeper~very aggressive spreader.

More like Ampelopsis arborea, peppervine.

Looks more like ampelopsis to me too. What's your zone? Although peppervine is not supposed to be hardy below zone seven, I can tell you it is. I had it in two locations on my property, purchased from an arboretum. In the more exposed location it died after two years, but in the protected one, planted near my brick house, it went nuts. Invasive isn't even the word I'd used. It was flame-thrower invasive. We cut it down and are still in the process of trying to kill them coming up quite a distance from the original. BTW we were considered zone five/six cusp but are now rated solid zone six.

I live in Tennessee which is zone 6. Any ideas for getting rid of this vine?

Follow it to its roots, cut it off and dig out the roots.