Is $\{T_a < T_b\} \perp \{T_b \leq T\}$ true ? $T_a =\inf\{t\geq 0 \mid B_t = a\}$

Question

I'm trying to check wether two sets related to the Brownian motion are or not independent.

Let $(B_t)$ be a Brownian motion and $a,b \in \mathbb{R}$, $a \neq b$.

Consider the first time when the Brownian motion gets to a certain point: $$T_x = \inf\{t\geq 0 | B_t = x\}, \quad x \in \mathbb{R}$$

Let $T \geq 0$. Are the event $\{T_a < T_b\}$ and the event $\{T_b < T\}$ independent?

Every help will be appreciated