Metamath Proof Explorer


Theorem brsigarn

Description: The Borel Algebra is a sigma-algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016)

Ref Expression
Assertion brsigarn 𝔅 ∈ ( sigAlgebra ‘ ℝ )

Proof

Step Hyp Ref Expression
1 fvex ( topGen ‘ ran (,) ) ∈ V
2 sigagensiga ( ( topGen ‘ ran (,) ) ∈ V → ( sigaGen ‘ ( topGen ‘ ran (,) ) ) ∈ ( sigAlgebra ‘ ( topGen ‘ ran (,) ) ) )
3 1 2 ax-mp ( sigaGen ‘ ( topGen ‘ ran (,) ) ) ∈ ( sigAlgebra ‘ ( topGen ‘ ran (,) ) )
4 df-brsiga 𝔅 = ( sigaGen ‘ ( topGen ‘ ran (,) ) )
5 uniretop ℝ = ( topGen ‘ ran (,) )
6 5 fveq2i ( sigAlgebra ‘ ℝ ) = ( sigAlgebra ‘ ( topGen ‘ ran (,) ) )
7 3 4 6 3eltr4i 𝔅 ∈ ( sigAlgebra ‘ ℝ )