Proving that probability measure is monotone

Question

I'm trying to understand a simple proof regarding probability (monotocity of a probability measure), i.e. proving that $A \subseteq B \to P(A) \leq P(B)$. Looking for an answer, I found ProofWiki's article.

I was able to get to $P(B) = P(A) + P(B-A)$, but I don't know how to conclude that $P(A) \leq P(B)$, knowing that $P(B-A) \geq 0$. Intuitively, I can see why it's true, but I don't know why I'm able to mathematically claim that. Can anyone help me clear this up?

Answer 1

$P(B-A) \geq 0$, or equivalently $P(A) + P(B-A) \geq P(A) $ by the measure axioms, so $P(B) = P(A) + P(B-A) \geq P(A)$.