For a continuous random variable X, $X\sim EXP(\theta) \iff$ $P(X > a + t | X > a) = P(X > t)$
Proof:
$P(X > a + t | X > a) = \dfrac{P(X > a + t \cap X >a)}{P(X > a)}$
$= \dfrac{P(X > a + t)}{P(X > a)}$
$= \dfrac{e^{-(a+t)/\theta}}{e^{-a/\theta}}$
$= P(X>t)$
Can someone explain to me why $P(X > a + t \cap X >a) = P(X > a + t)$