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 ∈ Prob → ( 𝑃 ‘ ∪ dom 𝑃 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		elprob	⊢ ( 𝑃 ∈ Prob ↔ ( 𝑃 ∈ ∪ ran measures ∧ ( 𝑃 ‘ ∪ dom 𝑃 ) = 1 ) )
2	1	simprbi	⊢ ( 𝑃 ∈ Prob → ( 𝑃 ‘ ∪ dom 𝑃 ) = 1 )