Implications of contemporaneous exogeneity

For some reason I don't feel confident, but in the context of time-series econometrics, does:


Where $u_t$ is the error term, and $x_t$ the matrix of all $x_{jt}$ for all explanatory variables j at time t.

Imply all of the following, via the law of iterated expectations?:
$E[u_t]=0$
$Cov[u_t,x_{jt}]=0 \forall j,t$
$E[u_t x_{jt}]=0 \forall j,t$