0.1 M sodium citrate in 10% ethanol and DNA solubility
The sodium ions will bind to and neutralize the (-) charge on the phospho backbone of DNA. This reduces hydrogen bonding sites for water, which then results in less solubility.

Edited for clarity and because I didn't answer the question the first time. Sorry about that. 

The extraction of DNA is from the interphase layer, which is a very small fraction of the original sample. The extraction of the interphase layer is then mixed with 100% EtOH (300 ul per 1.0 Trizol used in that sample). This should actually be 60% or more EtOH based on my experience using this method. 

That is the precipitation step. 

The fact that the wash has only 10% EtOH and 0.1M sodium citrate and that is still sufficient to keep DNA precipitated revolves around 0.1M sodium citrate is usually 0.3M sodium ions (sodium citrate can be mono-, di- or tri-sodium, but unless specified..sodium citrate implies tri-). Remember for the wash, you don't need to do a quick precipitation, you just need to prevent it from solubilizing again.

I am removing my comments that contained the same information.