Is there any possibility that our Universe was just born inside a Black Hole? If so, then mathematically prove that it happened and if it didn't happen, then please mathematically disprove it.
No. The reason is that black holes have a spacetime geometry that is very different from the universe.
The universe is known to have a nearly flat geometry, with perhaps some positive or negative curvature. Most importantly it is to a high degree isotropic: it looks the same in all directions. It is expanding, with more remote galaxies moving away faster from us than nearby galaxies.
Near a big black hole the geometry is very directional. The radial coordinate behaves very differently from the other two spatial directions: a spherical object even when falling freely is squeezed from the sides and pulled apart radially. When you are inside the event horizon this remains true, although the radial coordinate in some sense becomes like time: it is not possible to move towards larger $r$. The spaghettification still applies.
In a very big black hole these effects are milder than in a small one, but they still imply things very different from what we observe: a spatial anisotropy, objects moving towards each other rather than apart, and increasing curvature as we move towards the future.