We have mountains of evidence about singularities and how they work, and we have mountains of evidence that the "Big Bang" was the origin of the universe as we know it. But if compressing enough matter into a small space causes an inescapable singularity (i.e., a black hole), why does compressing all matter into a small space lead to a sudden expansion?
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It seems that the answer to my question is that there's a difference between the pre-inflation universe "singularity" and that of other types of singularities like a black hole. It wasn't a singularity as describes a black hole, it was a point in time where the scale of the universe was zero. All that exists didn't occupy a single point in space in the way matter collapses into a singularity in the universe we know, and the density of the universe was still homogenous. But the scale of it was reduced so much that the distance between everything was essentially zero, despite being an infinite universe, hence the big bang "happening everywhere at once" instead of at what we consider in our narrow view to be a single point.
As pointed out in the comments, this question has a very good explanation of how that works.
Edited with corrections suggested in the comments, thanks!
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The "explosion" wasn't caused by matter being under high pressure. It is thought that during the first few moments after the creation of the Universe, it went through a phase called "inflation", where there was no matter or radiation or anything else, only potential energy, which lasted $\sim10^{-34}$–$10^{-32}$ seconds and blew space up by a factor of at least $e^{60}$. See e.g. this post for more on inflation. When inflation ended, the Universe continued to expand but slowly decelerated due to the gravitational attraction of its matter. Recently, though, it started accelerating again due to something called dark energy.
