It is often helpful to plug in definitions; indeed, this is the whole point of clearly writing down definitions! If you are trying to prove two things are equal, find the appropriate definition of equality and plug it in.
We often indicate that we are using a definition by writing by definition, in other words, or i.e..
You can often save yourself some handwriting by expanding the outermost definitions first. For example, if you are proving , you could choose to expand any of the uses of the definition of the , of , or of ⊆. If you reduce or first, you will likely have to repeat its defintiion several times. If you expand first, on the other hand, you can immediately dive into the proof of without having to copy long formulas down.
