If you mean $(x_1,x_2)\not=(x_3,x_4)$, you can enforce this with:
$$10(x_1-1) + (x_2-1) \not= 10(x_3-1)+(x_4-1)$$
Equivalently:
$$10x_1 + x_2 \not= 10x_3+x_4 \tag1$$
If instead you meant the four disequalities you listed,
you can impose $$(x_1 - x_3)(x_1 - x_4)(x_2 - x_3)(x_2 - x_4)\not=0$$
Alternatively,
you might consider generating your four disequalities dynamically only if they are violated. In this case, $(1)$ is still valid but is only a relaxation.
Here's a MILP formulation. Let binary variable $y_{i,j}$ indicate whether $x_i=j$, and let binary variable $z_{g,j}$ indicate whether any variable in group $g$ is assigned value $j$. Then your constraints are
\begin{align}
\sum_j y_{i,j} &= 1 &&\text{for $i\in\{1,\dots,1000\}$} \\
\sum_j j y_{i,j} &= x_i &&\text{for $i\in\{1,\dots,1000\}$} \\
y_{i,j} &\le z_{g,j} &&\text{for $g\in\{1,2\}$, $i$ in group $g$, and $j\in\{1,\dots,10\}$} \\
\sum_g z_{g,j} &\le 1 &&\text{for $j\in\{1,\dots,10\}$}
\end{align}