We know from observational evidence that the universe is expanding today, so that it must have been smaller long ago. We can actually see this crowding effect by looking at very distant clusters of galaxies and notice that their constituent galaxies are more closely packed than typical clusters today.

Because the equations that describe Big Bang cosmology are 'scale free', you can extrapolate the scale, temperature and density of the universe back into time as far as you want. There is no known force that would have resisted this 'compression'.

By 1 second after the big band, the scale of the universe was 10 billion times smaller than it is today. Distant quasars located 10 billion light years away, were only 1 light year away after one second, but since light could only travel 1 light second, these clumps of matter were outside the visible universe at that time, which had a distance from out position of 1 light second. At this time, there are no forces that would have prevented the cosmos from being even smaller in the way that a spring eventually cant be compressed further. Here is roughly what this time looked like:

The equations let us go back even farther, again with the assumption that the expansion process was a continuous one, to a time that was even hotter and denser. Eventually, the separation between the farthest objects in our universe today, were less than the diameter of an atom.

Inflationary Big Bang cosmology lets us journey to a time 10^-35 seconds after the Big Bang when all that we can see today was crushed into a region of space about 10^-28 centimeters across, or about 100 trillion times smaller than the nucleus of an atom. Once again, we see no evidence that some force was available to prevent the universe from having been even smaller. Here is a sketch of what the universe would have been like around that time:

The point is that all current versions of Big Bang theory imply that the universe emerged from a very dense hot initial state, and that what we see as the entire visible universe today, out to 15 billion light years, was long ago crushed into a region smaller than a modern atom.

So, does this all add up to a compelling proof that the universe was once smaller than an atom? For astronomers and physicists it does.

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