It is not true in general that P(A|B) >= P(A). For example, suppose you are rolling a die that could be 6-sided (cube) or 4-sided (tetrahedron); then let A = “the die comes up showing a 6” and B = “the die is 4-sided”. We have P(A|B)=0, yet P(A) > 0 as long as P(B) < 1.
It is not true in general that P(A|B) >= P(A). For example, suppose you are rolling a die that could be 6-sided (cube) or 4-sided (tetrahedron); then let A = “the die comes up showing a 6” and B = “the die is 4-sided”. We have P(A|B)=0, yet P(A) > 0 as long as P(B) < 1.
He probably didn’t use standard notation here. I would read P(A|B) as P(A OR B) in this context.