A reference or proof should be given if it is not well known.
But for well known theorems it is not necessary.
For example one does not need to prove Pythagoras' theorem or give a reference to a proof every time one uses it. One only needs to say "by Pythagoras' theorem, ...."
There are 2 reasons Bertrand's postulate is called that and not "Bertrand's theorem".
Firstly although he first made the postulate in 1845, he wasn't the one to prove it. He did however give some numerical evidence with some examples. It is sometimes referred to as Tchebychev's theorem because he was the first one to prove it 5 years later. Nevertheless "Bertrand's postulate" is more commonly used to give credit to Bertrand because he came up with the idea, not Tchebychev.
The second reason is to distinguish it from another theorem which is called "Bertrand's theorem", which is something completely different related to mechanics.