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	$$⊢\bigcup {𝔅}_{ℝ}=ℝ$$

Proof

Step	Hyp	Ref	Expression
1		retop	$$⊢\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)\in \mathrm{Top}$$
2		unisg	$$⊢\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)\in \mathrm{Top}\to \bigcup 𝛔\left(\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)\right)=\bigcup \mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)$$
3	1 2	ax-mp	$$⊢\bigcup 𝛔\left(\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)\right)=\bigcup \mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)$$
4		df-brsiga	$$⊢{𝔅}_{ℝ}=𝛔\left(\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)\right)$$
5	4	unieqi	$$⊢\bigcup {𝔅}_{ℝ}=\bigcup 𝛔\left(\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)\right)$$
6		uniretop	$$⊢ℝ=\bigcup \mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)$$
7	3 5 6	3eqtr4i	$$⊢\bigcup {𝔅}_{ℝ}=ℝ$$