saying that the matrix has full rank means that it is of rank 2 - so there are two columns which are independent. you don't really know which 2, but for most proofs it doesn't matter. so you say without loss of generality, the first two columns are independent and therefore is of rank 2. the proof would probably work as well for the 1st and 3rd columns if they are independent too, etc.

by the way, you might have a 2x3 matrix that every two columns are independent, yet it must be of rank 2, for example