Impossible sums - proof 1

The problem Impossible Sums asks you to think about which numbers can be written as a sum of consecutive numbers, and which numbers cannot.

Here is a proof that the sum of consecutive numbers cannot be a power of $2$. 

The statements have been scrambled up. Can you rearrange them into their original order?

The first and last statements have been fixed in place, so you don't need to try and rearrange them.