For arbitrary events A, B, we have
${\displaystyle P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)\mathrm{.}}$ ...(1)
Since probability of an event is less then or equal to 1, we have
${\displaystyle P(A\cup B)\le 1}$ and so (1) gives
${\displaystyle P\left(A\right)+P\left(B\right)-P(A\cap B)\le 1}$
or ${\displaystyle P\left(A\right)+P\left(B\right)-1\le P(A\cap B)}$
or ${\displaystyle P(A\cap B)\ge P\left(A\right)+P\left(B\right)-1.}$