operators - $T bar{T}$ OPE

In page 157 of Di Francesco (Conformal Field Theory) it is said that the holomorphic and antiholomorphic components of the energy-momentum tensor have the trivial OPE

$T(z) \bar{T}(\bar{w}) \sim 0$.

I don't know if this was explained earlier in the book and I missed it or it is trivial and I don't see it. Do you have a proof?