Clarification about statements like $\forall n\in\mathbb{N}, \exists x_1,x_2,...,x_n \in {A},\forall i\in\{1,...,n-1\},x_i<x_{i+1}$

Question

When I see a statement like:

$\forall n\in\mathbb{N}, \exists x_1,x_2,...,x_n \in {A},\forall i\in\{1,...,n-1\},x_i<x_{i+1}$

The formal meaning is that:

$\forall n\in\mathbb{N}, \exists x:\{1,...,n\}\to A,\forall i\in\{1,...,n-1\},x(i)<x(i+1)$ ?

Thanks.