Description: Subclass relationship for class union. Theorem 61 of Suppes p. 39. (Contributed by NM, 22-Mar-1998) (Proof shortened by Andrew Salmon, 29-Jun-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | uniss | |- ( A C_ B -> U. A C_ U. B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssel | |- ( A C_ B -> ( y e. A -> y e. B ) ) |
|
| 2 | 1 | anim2d | |- ( A C_ B -> ( ( x e. y /\ y e. A ) -> ( x e. y /\ y e. B ) ) ) |
| 3 | 2 | eximdv | |- ( A C_ B -> ( E. y ( x e. y /\ y e. A ) -> E. y ( x e. y /\ y e. B ) ) ) |
| 4 | eluni | |- ( x e. U. A <-> E. y ( x e. y /\ y e. A ) ) |
|
| 5 | eluni | |- ( x e. U. B <-> E. y ( x e. y /\ y e. B ) ) |
|
| 6 | 3 4 5 | 3imtr4g | |- ( A C_ B -> ( x e. U. A -> x e. U. B ) ) |
| 7 | 6 | ssrdv | |- ( A C_ B -> U. A C_ U. B ) |