Deriving anti-commutation relation between creation/annihilation operators for Dirac fermions
Starting from Dirac fields:
$$\Psi(x) = \dfrac{1}{(2\pi)^{3/2}} \int \dfrac{d^3k}{\sqrt{2\omega_k}}\sum_r\left[ c_r(k)u_r(k)e^{-ikx}+d^\dagger_r(k)v_r(k)e^{-ikx} \right]_{k_0=\omega_k}$$
$$\Psi^\dagger(x) = \dfrac{1}{(2\pi)^{3/2}} \int \dfrac{d^3k}{\sqrt{2\omega_k}}\sum_r\left[d_r(k)v^\dagger_r(k)e^{-ikx} + c^\dagger_r(k)u^\dagger_r(k)e^{ikx}\right]_{k_0=\omega_k} $$
where $\omega_k = \sqrt{\vec{k}^2+m^2}$.
The canonical quatization condition reads:
$$ \begin{cases} {\Psi_\alpha(x), \Psi^\dagger_\beta(y)}t = \delta^{(3)}(\vec{x}-\vec{y})\ \ \delta{\alpha\beta}\ {\Psi_\alpha(x), \Psi_\beta(y)}t = 0\ {\Psi^\dagger\alpha(x), \Psi^\dagger_\beta(y)}_t = 0\ \end{cases} $$
In order to derive the quantization condition for the creation/annihilation operators I have to rewrite $c,c^\dagger,d,d^\dagger$ in terms of $\Psi$ and $\Psi^\dagger$.
For instance in order to derive the canonical quantization condition between $c,c^\dagger$ I can rewrite them as:
$$ c_r(k) = \dfrac{1}{\sqrt{2\pi}^{3}} \int \dfrac{d^3x}{\sqrt{2\omega_k}} u_r^\dagger(k) \Psi(x) e^{ikx} $$
$$ c^\dagger_s(p) = \dfrac{1}{\sqrt{2\pi}^{3}} \int \dfrac{d^3y}{\sqrt{2\omega_p}} \Psi^\dagger(y) u_s(p) e^{-ipy} $$
and then explicitly calculate the anti-commutator:
$$ \begin{split} {c_r(k), c^\dagger_s(p)}_t &= \dfrac{1}{(2\pi)^3} \int \dfrac{d^3xd^3y}{\sqrt{2\omega_k 2\omega_p}} \left[ u_r^\dagger(k) \Psi(x)\Psi^\dagger(y) u_s(p) + \Psi^\dagger(y) u_s(p)u_r^\dagger(k) \Psi(x) \right]e^{i(kx-py)}\ &= \dfrac{1}{(2\pi)^3} \int \dfrac{d^3xd^3y}{\sqrt{2\omega_k 2\omega_p}} \left[ u_r^\dagger(k) {\Psi(x), \Psi^\dagger(y)} u_s(p)\right]e^{i(kx-py)}\ &= \cdots \end{split} $$
But here I miss something: I don't understand why I can swap $u_s(p)$ and $u_r^\dagger(k)$ in the second term in order to recover the anti-commutator between $\Psi$ and $\Psi^\dagger$. 281