I'm trying to translate the following statement to first order set theory:
$$w \subseteq u\times v$$
I already have $u\times v$ defined to be the following:
$$u\times v=\{(x,y)\mid x\in u\land y\in v\}$$
How do I incorporate $w$ and the subset symbol?
HINT: first think about how you would write "$w\subseteq a$." Remember that this means "every element of $w$ is an element of $a$" - do you see how to express this in the first-order language of set theory?
Now, folding in "$u\times v$" will take a bit of work. What you want to write down is $$\mbox{All elements of $w$ are ordered pairs in $u\times v$}.$$ This will involve writing a first-order formula expressing that a set $x$ is an element of $u\times v$ (you need to figure out how to write "$x$ is an ordered pair," "the first coordinate of $x$ is in $u$," and "the second coordinate of $x$ is in $v$" in the first-order language of set theory). Note that ordered pairs are defined in terms of "$\in$" (e.g. via the Kuratowski definition), so the definition you've given above of $u\times v$ isn't really "done".