Coupling of Bernoulli variables by Monotone coupling

Let $$X$$ and $$Y$$ be Bernoulli random variables with parameters $$0\leq q< r\leq 1$$ respectively. That is $$P(X=0)=1-q=1-P(X=1)$$ and similarly for $$Y$$. To pick up a uniform random variable $$U$$ at $$[0,1]$$. Let $$X'=\mathbf{1}_{U\leq q}$$ and $$Y'=\mathbf{1}_{U\leq r}$$.

Why we can get

$$\mathbb{P}((X', Y')=(0, 0))=1-r$$ $$\mathbb{P}((X', Y')=(0, 1))=r-q$$ $$\mathbb{P}((X', Y')=(1, 0))=0$$ $$\mathbb{P}((X', Y')=(1, 1))=q?$$

$$\mathbb{P}((X', Y')=(0, 0))=\mathbb{P}(U>q, U>r) \overset{r>q}{=} \mathbb{P}(U>r)=1-r.$$ $$\mathbb{P}((X', Y')=(0, 1))=\mathbb{P}(U>q, U\leq r) = \mathbb{P}(q Try to consider the last two equalities in the same manner.