1. it is not true that (for all hats h in {liar's hats}, h is green)

2. by the negation rule, there exists a hat h in {liar's hats}, such that it is not true that h is green.

3. there exists a hat in {liar's hats}

This is really basic first order logic guys.

"For all x in {}, P(x)" is always trivially true without regard for P.

Similarly, "there exists an x in {}, P(x)" is always trivially false.

https://en.wikipedia.org/wiki/Vacuous_truth