  where each Qk is ∀ or ∃ and φi is a (possibly infinitary) conjunction or disjunction of formulas of the form xk = xl or xk ≠ xl. Since each σi is a sentence, there are only finitely many variables in each φi, and it is easy to see that each φi is then equivalent to a first-order formula. Accordingly each σi may be taken to be a first-order sentence. Since σ is assumed to be equivalent to the conjunction of the σi, it follows that σ and the set Δ = {σi : i ∈ I} have the same models. But obviously σ, and hence also Δ, have models of all finite cardinalities; the compactness theorem for sets of first-order sentences now implies that Δ, and hence also σ, has an infinite model, contradicting the definition of σ. 