I was reading the posted to solutions to one of the questions on a probability midterm and couldn't figure out how to justify one of the steps.
Let $\{B_t\}_{t\geq 0}$ be a Brownian motion and $Z(\omega) = \{t\geq 0 : B_t(\omega) = 0\}$ its zero set. The question is to compute $$ \mathbb P[Z \cap [1,2] = \emptyset]. $$ The very first step is to notice that the conditional probability $$ \mathbb P[Z \cap [1,2] \mid B_1 = x] = \mathbb P[\sup_{t \leq 1} B_t < |x|], $$ but I'm not seeing this equality. Any help would be appreciated!