$\alpha$ condition and choice structure
A choice structure can be whatever it wants to be. A choice structure that is rationalizable by a preference relation must (among other things) satisfy Sen's $\alpha$. 

So if you want to show that a rationalizable choice structure satisfies $\alpha$ then you can show that any that satisfies WARP will also satisfy $
\alpha$. 

I recommend Kreps or the UPitt notes for more info on the relationship between choice structures and preference. 

http://www.pitt.edu/~luca/ECON2100/lecture_02.pdf