To start off, you didn't prove aliens exist. To put it plainly, to prove something is to actually show physical evidence of its existence. You simply used logic to attempt to prove something, but never did prove that aliens exist. But if you manage to get a hold of a selfie with an alien, then feel free to state that you have proved the existence of aliens. Also aliens have been proven true long before you posted this, with actual evidence. Problem is, the common public imagines an alien as a biped creature, with large black eyes, long fingers, and whatever your strange imagination makes you think of an alien, but in reality aliens can be just bacteria living in water. NASA has found plenty of planets with data to prove that water is plentiful on these strange foreign planets, and that there's enough heat to sustain life for microbes.

Extraterrestrial life, also called alien life (or, if it is a sentient or relatively complex individual, an "extraterrestrial" or "alien"), is life that does not originate from Earth.

It's statistically incredibly likely alien life exists, but whether anything more than microscopic life exists is basically entirely unknown. We cannot prove with certainty even simple flora or fauna exists on other worlds. There's a popular concept known as "The Great Filter" which suggests there are one or more large-scale events that stop life in it's tracks. The hope is that those events are "behind" us and we survived them.

The Fermi Paradox is another relevant theory and the reason for the existence of the Great Filter theory. Essentially, it's incredibly odd how we've seen no trace of intelligent alien life despite the universe being as vast and complex as it is. There are many, many potential explanations for this theory and it's all quite fascinating. There's, of course, The Great Filter theory, which suggests we're one of the few to pass, or one of many that has yet to encounter the "filter." There's a theory that there may be powerful sapient races out there or something dangerous tracking communication, and, as such, all civilizations try to hide themselves. There's also the concept that we just live "in the country," as it were, far away from civilized space. Our SETI technology is very primitive compared to what a race capable of interstellar flight would use. There are plenty of other theories on the topic; too many to recount here.