Example (Raster A - Raster B)
Raster A
Min 5
Max 10
Raster B
Min 1
Max 10
The Min value of A is larger than that of B, per your criteria.
Lets say at a given pixel location, Raster A is value of 6, and Raster B is value of 10. This would result in a value of -4.
It doesn't matter what the minimum values are. It matters what the values are at the pixel locations being subtracted.
Regards,
Eric