We know that the lower atmosphere has high pressure and as we go up, the pressure decreases, if it's so then why doesn't all gases fly up into the upper atmosphere from the lower following Bernoulli's theorem? I do expect that gravitational effect on gases isn't that worth notable. Do correct me for my mistake if it exists!
If you have ever swum to the bottom of a swimming pool you'll know that in water the pressure increases as you go deeper. At a depth of about 10 metres the pressure is twice what it is at the surface, but the water 10 metres down doesn't burst up to the surface because it is held down by the weight of water above it. In fact the increase of pressure with depth is exactly the weight of water above.
Exactly the same is true of the atmosphere. The pressure at ground level is 101,325 Pa because each square metre of the ground has about 10,329 kg of air above it (10329 kg times the acceleration due to gravity 9.81 m/sec$^2$ = 101325 Pa). If you could magically remove the 100 km or so of atmosphere that's above some patch of air at ground level that air would indeed immediately expand upwards.
Incidentally, Bernoulli's principle is unrelated to the problem.