proof of compactness theorem for first order logic

The theorem states that if a set of sentences of a first-order language $L$ is inconsistent, then some finite subset of it is inconsistent. Suppose $\mathrm{\Delta }\subseteq L$ is inconsistent. Then by definition $\mathrm{\Delta }⊢⟂$, i.e. there is a formal proof of “false” using only assumptions from $\mathrm{\Delta }$. Formal proofs are finite objects, so let $\mathrm{\Gamma }$ collect all the formulas of $\mathrm{\Delta }$ that are used in the proof.