That
(1) Ext$^n_\mathbb{Z}(M, \mathbb{Q})=0$ for every module $M$
follows easily from the fact that
(2) $\mathbb{Q}$ is injective.
However, the only proof I have seen of the injectivity of $\mathbb{Q}$ relies on Baer's Criterion. While the proof of Baer's Criterion is not difficult, it seems stronger than the injectivity of $\mathbb{Q}$ (for example, the proof uses Zorn's Lemma). Is there a different (and preferably simpler) proof of (1) or (2)?