Unfortunately, this is wrong. What you probably meant is that for all v, w ∈ V, we have v - w ∈ V. What you wrote is not enough. For example, take V = {0, 1, -1} as a subset of say ℚ (regarded as a vector space over itself).

Also the terminology "negative vector" is slightly imprecise. Better to say "opposite".

Finally, saying "properties 3 and 4" doesn't really make much sense. I guess those refer to your textbook, but the numbering is not universal, so most people wouldn't know which properties you're talking about.

random math lesson amid typing practice absolutely loved it

Hell yeah.

wuut?