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 ) ) = (/) |