No, no no! You really want to keep the rootball intact. You'll likely kill your seedlings if you strip the soil from the roots.
The way I do it in the Solo cups is super easy.
First I get the new pot prepared, and I use the cup, with the seedling still in it, to make the new hole (or at least gauge the depth and approximate circumference) or at least an indentation as a guide.
Then I grab the cup so that the seedling is between my fingers in such a way that when I turn it upside down all the dirt will be resting in my palm, with the seedling hanging underneath.
Then, carefully! squeeze the cup so that the the rootball and soil slide out, hopefully it's all stuck together in one piece (make sure the soil is moist so it doesn't just fall apart).
Now, carefully flip it over into the new hole. Some soil might come off. If you need to bring the whole thing up, lean it over and push some soil underneath, continuing all around til it's where you want its final growing place to be.
Make sure it's centered where you want it, and upright.
Now, starting from the bottom up, firm the soil down. Make sure the soil level to the seedling as close to the original level as possible.
Don't try to use less soil in the new pots, fill them up to within an inch or two.