Theorem probtot

Description: The probability of the universe set is 1. Second axiom of Kolmogorov. (Contributed by Thierry Arnoux, 8-Dec-2016)

	Ref	Expression
Assertion	probtot	$$⊢P\in \mathrm{Prob}\to P\left(\bigcup \mathrm{dom}P\right)=1$$

Proof

Step	Hyp	Ref	Expression
1		elprob	$$⊢P\in \mathrm{Prob}\leftrightarrow \left(P\in \bigcup \mathrm{ran}\mathrm{measures}\wedge P\left(\bigcup \mathrm{dom}P\right)=1\right)$$
2	1	simprbi	$$⊢P\in \mathrm{Prob}\to P\left(\bigcup \mathrm{dom}P\right)=1$$