For a sequence of random variables $\left\{ X_1, \ldots, X_{N} \right\}$, not independent, or identically distributed. From Nagaraja and David The CDFs of the order statistics $\left\{ X_{\left( 1 \right)}, \ldots, X_{\left( N \right)} \right\}$, can be found recursively according to the formula $$\sum_{i=1}^{n} F_{r:n-1}^{\left( i \right)} \left( x \right) = r F_{r+1:n} \left( x \right) + (n-r)F_{r:n}\left( x \right)$$
If the following equality is true: $$F_{n:n}\left( x \right) = P\left( X_1 \leq x, \ldots, X_N \leq x \right)$$ then every order statistic can always be found recursively from the joint distribution of the sequence of random variables $\left\{ X_1, \ldots, X_N \right\}$.
However I'm unconvinced that the equality always holds, because the joint distribution of an arbitrary sequence of random variables can take arbitrary values. And so I'm not sure that exchangeability of these random variables will always hold: ie.: $$P\left( X_1 \leq x, \ldots, X_n \leq x \right) \neq P\left( X_{\sigma\left(1\right)} \leq x, \ldots, X_{\sigma\left( N \right)} \leq x \right)$$
In the general case it's clear that: $$\sum_{i=r}^{n} {n \choose i} P\left(X_1 \leq x, \ldots, X_i \leq x, X_{i+1} \geq x, X_n \geq x \right) \neq P\left( X_{\left( r \right)} \leq x \right)$$ because the lack of exchangeability between random variables $\left(X_1, \ldots, X_N \right)$ prevents us being able to take the simple sum of scaled joint cumulative density functions.
The formula given in Nagaraja and Davids book for the general case is: $$\sum_{j=r}^{n}(-1)^{j-r} {j-1 \choose r-j} \sum_{1 \leq i_1 < \cdots < i_j \leq n} \text{Pr}\left( A_{i_1}, \cdots, A_{i_j} \right)$$ where the sum $$S_j = \sum_{1 \leq i_1 < \cdots < i_j \leq n} \text{Pr}\left( A_{i_1}, \ldots, A_{i_j} \right)$$ is taken over all permutations of the events $$\left\{ A_{i_1}, \ldots, A_{i_j} \right\} \subset \left\{ A_1, \ldots, A_n \right\}$$
Is there a proof that the following equality always holds? $$F_{n:n}\left(x\right) = P\left( X_1 \leq x, \ldots, X_N \leq x \right)$$
Or maybe a counter example? that prevents me from just using the joint probability of $\left\{ X_1, \ldots, X_N \right\}$ to derive all of the order statistics if I know it?