True or False: $P(A \cup B \cup C) \geq P(A | B \cup C) P(B) P(B \cup C)$ I am trying to determine if
$$P(A \cup B \cup C) \geq P(A | B \cup C) P(B) P(B \cup C)$$
I tried
$$\begin{align}
P(A \cup B \cup C) \geq P(A | B \cup C) P(B) P(B \cup C) = P(A \cap (B \cup C) ) \cdot P(B)
\end{align}$$
which is not leading to anywhere. I feel like I am missing something really obvious.
Thank you.

$P(A \cup B \cup C) \ge P(B)$ because $ B \subseteq A \cup B \cup C$.  Multiplying the right side by a probability and a conditional probability can't make it bigger.
