Why use $\forall$ instead of $\in$? So, here are two ways to say what I interpret as the same statement:

$f_i(x,y)\geq0 \hspace{0.85cm} \forall i \{0,1,2\}$

which implies that $f_0(x,y) \geq0$ and $f_1(x,y)\geq0$ and $f_2(x,y)\geq0$
but doesn't

$f_i(x,y)\geq0 \hspace{0.85cm}\{i\in\mathbb{Z}|i\in[0,2]\}$

imply the same thing?
Is there another reason why these different notations are used, besides the fact that the one consumes less space than the other? Apologies if one of these notations falls into a specific category of mathematics without my knowledge. I am not fully taught (evidently).
Any responses are appreciated.

Actually, in formal mathematics, the condition should come first:
$$ \forall i \in \{0,1,2\} \; f_i(x,y) \ge 0 $$
