Proof: $P(\varnothing)=0$

Question

To prove(using the three probability axioms): $$P(\varnothing)=0$$ Is this method correct?

Let $A$ be an event such that $A=\Omega$. Then $A^\complement=\varnothing$, $$P(A)+P(A^\complement)=1$$ $$P(\Omega)+P(\varnothing)=1\tag2$$ $$P(\varnothing)=0$$

Please, post any proof that is better/correct.

Answer 1

The method that you use is correct.

You can go for:$$1=P(S)=P(S\cup\varnothing)=P(S)+P(\varnothing)=1+P(\varnothing)$$implying that $P(\varnothing)=0$.

Here the first equality is a consequence of the axioma saying that $P(S)=1$ and the third equality is a consequence of the axioma saying that $P(A\cup B)=P(A)+P(B)$ is implied by $A\cap B=\varnothing$.

Actually a proof that is better does not exist.