Well, stylet is very resistant, in reality it is made by more layers of tissue: Labrum, labella etc. (the complex of them is also called fascicle) and in its terminal part is very rigid (to pierce) while in the rest is more elastic.
Also, it is designed to be similar to a needle, so that resistance by the host skin is reduced to minimum thus there is no risk of breaking.
It is like a needle... a very thin and sharp needle wouldn't break even it was made of crystal, due to the small resistance opposed. Plus, human skin is not as resistant as animals' one, so it is easy for the mosquito to pierce.
Just think that engineers are designing bionic needles based on mosquitoes (The Insect Fascicle Morphology Research and Bionic Needle Pierced Mechanical Mechanism Analysis by Xin et al.)