Theorem unibrsiga

Description: The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017)

	Ref	Expression
Assertion	unibrsiga	⊢ ∪ 𝔅ℝ = ℝ

Proof

Step	Hyp	Ref	Expression
1		retop	⊢ ( topGen ‘ ran (,) ) ∈ Top
2		unisg	⊢ ( ( topGen ‘ ran (,) ) ∈ Top → ∪ ( sigaGen ‘ ( topGen ‘ ran (,) ) ) = ∪ ( topGen ‘ ran (,) ) )
3	1 2	ax-mp	⊢ ∪ ( sigaGen ‘ ( topGen ‘ ran (,) ) ) = ∪ ( topGen ‘ ran (,) )
4		df-brsiga	⊢ 𝔅ℝ = ( sigaGen ‘ ( topGen ‘ ran (,) ) )
5	4	unieqi	⊢ ∪ 𝔅ℝ = ∪ ( sigaGen ‘ ( topGen ‘ ran (,) ) )
6		uniretop	⊢ ℝ = ∪ ( topGen ‘ ran (,) )
7	3 5 6	3eqtr4i	⊢ ∪ 𝔅ℝ = ℝ