Generalized delta-system lemma ($|\mathcal{A}|=\theta ⟹|\mathcal{B}|=\theta$)

If $\mu <\theta =\mathrm{cf}(\theta )$ are infinite cardinals and $|[\alpha {]}^{<\mu }|<\theta$ for every $\alpha <\theta$, every $\theta$-sized family of sets of size less than $\mu$ has a $\theta$-sized delta-system subfamily.