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After understading the argument such as above, I set my work to show the electric field inside conductor is zero(Given total charge inside of the conductor but not the surface is zero) rather mathematically and got nowhere. Any help or advise will be appreciated.

If I already know total charge inside of the conductor is zero, from Gauss's law it follows that Electric field inside of conductor must be constant value. So it means that it must be either zero or non-zero constant value. Now, how can I show that zero is the only value possible?