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It seems the proof is non-trivial. The claim appears (with proof) as Theorem 2.4.12 in [Hovey, Model categories] and as part of Theorem 17.2.2 in [May and Ponto, More concise algebraic topology]. It is also asserted (without proof) as Lemma 9.4 in [Goerss and Jardine, Simplicial homotopy theory, Ch. I].

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