HMMT November 2022 U14

Real numbers $x$ and $y$ satisfy the following equations:

$$\begin{aligned} x & =\log _{10}\left(10^{y-1}+1\right)-1 \\ y & =\log _{10}\left(10^{x}+1\right)-1 \end{aligned}$$

Compute $10^{x-y}$.