Hello, I am having troubling proving the antennas of the Mandelbrot Set have empty interior.

I have seen it done for a none general point on an antenna (not rigorously), for instance let then it is easy to show any point of the form for escapes to but I don't know how to start proving this for a general point, and without using the above method, which obviously doesn't suffice as a proof.

I realise this point isn't on an antennae, I was just giving an example of how I would "show" a point has 0-thickness.