Can emotional intelligence be proven to exist?
The thing that we call IQ (or g) is a set of cognitive abilities in pattern recognition, problem solving, puzzles, symbol manipulation, and so forth. Insofar as "something" is measured by these kinds of tests, then IQ/g can be said to exist.
Similarly, the set of traits and behaviors collected under the umbrella of "emotional intelligence" are a grab bag of self-regulation, resilience, emotional self-control, insight into one's own mind, ability to form a working model of the emotions of others, ability to "read" the emotions of others, and so forth. These are less well defined than the abilities grouped under IQ/g but insofar as they could be measured through well-designed tests, then emotional intelligence can be said to exist.
Alternatively, there could be a functional or "effectiveness based" definition of emotional intelligence, even if it were less quantitative than IQ/g.
Arguing over definitions and labels is usually unproductive. I think in this case, it does dilute the idea of intelligence (being able to reason abstractly), while at the same time, increases the emphasis on high intelligence.
We'd all be better off just defocusing on intelligence, as a: there's little control you have over your intelligence, and b: it matters far, far, less than what some of us programming types might think, and c: focusing on an innate skill tends to come at a cost of not focusing on perseverance and work ethic, which matter far more.
Personally, I wish there'd been less emphasis on how smart I was as a kid; perhaps I'd have more discipline. But alas, thinking about algorithms causes far more short-term reward than, say, cold calling, yet the latter is far more likely to generate success for me.