Notice how the egg vibrates as the heated air escapes around it, and how it gets smoothly pushed into the bottle by atmospheric ... air pressure is greater than the pressure inside the cooled bottle.

Bottles of soda are pressurized above atmospheric pressure ... dioxide is added to a sealed bottle or can containing water, ...

so as water vapor increases, the density of air decreases. Temperature also changes the density of air. As air gets warmer it expands and becomes less dense, causing atmospheric pressure to fall.