  We present systematic measurements of the mechanical properties of few unit
cell (UC) thick exfoliated crystals of a high-T$_c$ cuprate superconductor
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. We determine the elastic properties of
these crystals by deformation using an atomic force microscope (AFM) at room
temperature. With the spatial measurements of local compliance and their
detailed modeling, we independently determine the Young's modulus of rigidity
and the pre-stress. The Young's modulus of rigidity is found to be in the range
of 22 GPa to 30 GPa for flakes with thickness from 5 UC to 18 UC. The
pre-stress spreads over the range of 5 MPa - 46 MPa, indicating a run-to-run
variation during the exfoliation process. The determination of Young's modulus
of rigidity for thin flakes is further verified from recently reported buckling
technique.
