I was responsible for killing that big vine that betsy mentioned. My hands were shaking as I sawed through it with my bow saw. I just took out a 2 foot section at the bottom and left it. You will need to be careful because where the mother vine is; the soil will be saturated with the oil and other vines. Not to scare you but you will see the blade of your saw turn black as the urushiol oil oxidizes with air. That huge hairy vine will need to be pried out with something because even after you cut both sections it will be stuck to your tree with all that hair. I just used the tip of my ax to pry it out. Dispose of this segment and wash your tools and yourself immediately. I just use good pressure and lots of water for the tools. Assume you have oil on you. Return and spray the other ground plants and the bottom vine with round-up or similar.

Don't use anything that will cause the oil to become airborne such as a chain saw. As you noticed on your vine, the urushiol oil is black when exposed to air. It is still a very very potent form. I have been told that even a pin head of this stuff would cause a systemic reaction.