Theorem sssigagen

Description: A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017)

	Ref	Expression
Assertion	sssigagen	$$⊢A\in V\to A\subseteq 𝛔\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssintub	$$⊢A\subseteq \bigcap \left\{s\in \mathrm{sigAlgebra}\left(\bigcup A\right)|A\subseteq s\right\}$$
2		sigagenval	$$⊢A\in V\to 𝛔\left(A\right)=\bigcap \left\{s\in \mathrm{sigAlgebra}\left(\bigcup A\right)|A\subseteq s\right\}$$
3	1 2	sseqtrrid	$$⊢A\in V\to A\subseteq 𝛔\left(A\right)$$