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.