Metamath Proof Explorer

Theorem unveldomd

Description: The universe is an element of the domain of the probability, the universe (entire probability space) being U. dom P in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017)

		Ref	Expression
	Hypothesis	unveldomd.1	$⊢ φ \to P \in Prob$
	Assertion	unveldomd	$⊢ φ \to ⋃ dom P \in dom P$

Proof

Step	Hyp	Ref	Expression
1		unveldomd.1	$⊢ φ \to P \in Prob$
2		domprobsiga	$⊢ P \in Prob \to dom P \in ⋃ ran sigAlgebra$
3		sgon	$⊢ dom P \in ⋃ ran sigAlgebra \to dom P \in sigAlgebra (⋃ dom P)$
4		baselsiga	$⊢ dom P \in sigAlgebra (⋃ dom P) \to ⋃ dom P \in dom P$
5	1 2 3 4	4syl	$⊢ φ \to ⋃ dom P \in dom P$