Notation for exists two elements in a set with properties

Question

I'd like to say: for any x in set X, if x is colorful, there must be t1 and t2, both in set T, such that t1 < t2 and green(x,t1) and red(x,t2). I believe this is the correct notation, but I'm not sure:

$\forall x \in X: colorful(x) \Rightarrow \exists t_1 \in T, t_2 \in T: t_1 < t_2 \wedge green(x,t_1) \wedge red(x,t_2)$

My questions:

• Did I use the colon properly for 'such that'? It seems funny to have two colons, but it seems even worse to drop the second colon or replace it with and And symbol
• Can I combine t1 and t2 to something like $\exists t1,t2 \in T$

Answer 1

If I were going to write it out symbolically in this fashion, I’d write

$$\forall x\in X\Big(\operatorname{colorful}(x)\to\exists t_1,t_2\in T\big(t_1<t_2\land\operatorname{green}(x,t_1)\land\operatorname{red}(x,t_2)\big)\Big)\;.$$

In general, though, I prefer something with a few more words:

If $x\in X$ is colorful, there are $t_1,t_2\in T$ such that $t_1<t_2$, $\operatorname{green}(x,t_1)$, and $\operatorname{red}(x,t_2)$.