What proof is there that the Big Bang Theory is false?
The Big Bang is the most widely accepted theory of the development of the universe from a stage where it was very small and hot.
All our observations are consistent with it.
There are some things it does not explain. For example, it does not describe how the universe came into being. Neither does it explain why there is much more matter than antimatter.
It is subject to refinement as a theory, but there have been no observations which refute it.