Description: An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | elsigass | |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> A C_ U. S ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgon | |- ( S e. U. ran sigAlgebra -> S e. ( sigAlgebra ` U. S ) ) |
|
2 | sigasspw | |- ( S e. ( sigAlgebra ` U. S ) -> S C_ ~P U. S ) |
|
3 | 1 2 | syl | |- ( S e. U. ran sigAlgebra -> S C_ ~P U. S ) |
4 | 3 | sselda | |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> A e. ~P U. S ) |
5 | 4 | elpwid | |- ( ( S e. U. ran sigAlgebra /\ A e. S ) -> A C_ U. S ) |