Description: The Borel Algebra is a sigma-algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | brsigarn | ⊢ 𝔅ℝ ∈ ( sigAlgebra ‘ ℝ ) |
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 | ⊢ 𝔅ℝ = ( sigaGen ‘ ( topGen ‘ ran (,) ) ) | |
5 | uniretop | ⊢ ℝ = ∪ ( topGen ‘ ran (,) ) | |
6 | 5 | fveq2i | ⊢ ( sigAlgebra ‘ ℝ ) = ( sigAlgebra ‘ ∪ ( topGen ‘ ran (,) ) ) |
7 | 3 4 6 | 3eltr4i | ⊢ 𝔅ℝ ∈ ( sigAlgebra ‘ ℝ ) |