Transitivity of almost surely

Question

Suppose an event $A$ occurs with probability $1$, and given that $A$ occured, event $B$ occurs with probability $1$. How does one formally prove that with probability $1$, event $B$ occurs? I understand what the statements mean , but I am struggling with proving it formally. It came up in the context of recurrent states and communication classes.

Answer 1

$P(B) = P(B|A)P(A) + P(B|\tilde A)P( \tilde A)$

but if you know that $P(A)=1$ then $P(\tilde A)=0$ and so this simplifies to

$P(B) = P(B|A) = 1$