Does this outline of a proof work?

Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to show that they are not isometric.

One way to distinguish the two spaces is their sectional curvature. I think I have shown that the sectional curvature of the Bergman metric of the ball is constant and negative, whereas the sectional curvature of the bidisc is nonpositive non constant. For example in the plane generated by the vectors $\langle 1,0 \rangle$ and $\langle 0,1 \rangle$ the section curvature is 0, but in the plane generated by $\langle 1,0\rangle$ and $\langle i,0 \rangle $ the sectional curvature is negative.

Is this true? Is there anything subtle I might have missed? I have seen a lot of pretty convoluted proofs of this fact, and I would think that this basic outline would be recorded in print somewhere if it is true, but I cannot seem to find it.