Hey downthesun01.
The general proof involves using the definitions (in continuous and discrete probability spaces) and basically "integrating out" the Y random variable after you integrate out the X random variable for a particular value.
Law of total expectation - Wikipedia, the free encyclopedia
For your particular case take the joint distribution, integrate out the x variable and then take the resulting function (which will depend on y) and integrate out the y variable.
Also note that when you integrate out you need to multiply the PDF by the variable (since E[X] = Integral x*f(x)dx for some RV x).