Joint pdf of two uniform random variables on a unit line segment

Question

Let $X$ be a standard uniform random variable, define $Y=1-X$. Then supposedly $X$ and $Y$ are uniform over a 1-simplex, so their joint distribution should be Dirichlet of order $K=2$, and $\alpha_1=\alpha_2=1$ . Hence, their joint pdf is $$f(x,y)=1\quad\text{if }x,y\in[0,1]\land x+y=1.$$ My question is, how do I derive the joint pdf from definition? A naive attempt is to notice the joint cdf $$F(x,y)=P(X\leq x,1-X\leq y)=P(1-y\leq X\leq x).$$ But then taking cross derivatives of $F$ seems yield $0$ everywhere...