Description: The intersection and class difference of a class with another class are disjoint. With inundif , this shows that such intersection and class difference partition the class A . (Contributed by Thierry Arnoux, 13-Sep-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | inindif | |- ( ( A i^i C ) i^i ( A \ C ) ) = (/) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 | |- ( A i^i C ) C_ C |
|
2 | ssinss1 | |- ( ( A i^i C ) C_ C -> ( ( A i^i C ) i^i A ) C_ C ) |
|
3 | 1 2 | ax-mp | |- ( ( A i^i C ) i^i A ) C_ C |
4 | inssdif0 | |- ( ( ( A i^i C ) i^i A ) C_ C <-> ( ( A i^i C ) i^i ( A \ C ) ) = (/) ) |
|
5 | 3 4 | mpbi | |- ( ( A i^i C ) i^i ( A \ C ) ) = (/) |