Description: Transitivity of subclass relationship. Exercise 5 of TakeutiZaring p. 17. (Contributed by NM, 24-Jun-1993) (Proof shortened by Andrew Salmon, 14-Jun-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sstr2 | |- ( A C_ B -> ( B C_ C -> A C_ C ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssel | |- ( A C_ B -> ( x e. A -> x e. B ) ) |
|
| 2 | 1 | imim1d | |- ( A C_ B -> ( ( x e. B -> x e. C ) -> ( x e. A -> x e. C ) ) ) |
| 3 | 2 | alimdv | |- ( A C_ B -> ( A. x ( x e. B -> x e. C ) -> A. x ( x e. A -> x e. C ) ) ) |
| 4 | dfss2 | |- ( B C_ C <-> A. x ( x e. B -> x e. C ) ) |
|
| 5 | dfss2 | |- ( A C_ C <-> A. x ( x e. A -> x e. C ) ) |
|
| 6 | 3 4 5 | 3imtr4g | |- ( A C_ B -> ( B C_ C -> A C_ C ) ) |