Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)
Ref | Expression | ||
---|---|---|---|
Assertion | iun0 | |- U_ x e. A (/) = (/) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel | |- -. y e. (/) |
|
2 | 1 | a1i | |- ( x e. A -> -. y e. (/) ) |
3 | 2 | nrex | |- -. E. x e. A y e. (/) |
4 | eliun | |- ( y e. U_ x e. A (/) <-> E. x e. A y e. (/) ) |
|
5 | 3 4 | mtbir | |- -. y e. U_ x e. A (/) |
6 | 5 | nel0 | |- U_ x e. A (/) = (/) |