The problem is: Suppose S is a satisfiable set of formulas, and G is a subset of S, and for every formula t, either t in G or ~t in G. Show S is a subset of G.

Proof: Since S is satisfiable, we have that S is consistent. Suppose t in S. Since S is consistent and G is a subset of S, we have that ~t not in G. So t in G.

EDIT after I realized (spurred by emakarov) that I didn't mean 'consistent' in the above:

Proof: Since S is satisfiable, we have that there is no formula t such that both t and ~t are in S. Suppose t in S. Since there is no formula t such that both t and ~t are in S, and G is a subset of S, we have that ~t not in G. So t in G.

The original proof is still correct, but the edited version is more economical since it does not require the soundness theorem.