Description: The union of the empty set is the empty set. Theorem 8.7 of Quine p. 54. (Contributed by NM, 16-Sep-1993) Remove use of ax-nul . (Revised by Eric Schmidt, 4-Apr-2007) Avoid ax-11 . (Revised by TM, 1-Feb-2026)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | uni0 | |- U. (/) = (/) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | noel | |- -. y e. (/) |
|
| 2 | 1 | intnan | |- -. ( x e. y /\ y e. (/) ) |
| 3 | 2 | nex | |- -. E. y ( x e. y /\ y e. (/) ) |
| 4 | eluni | |- ( x e. U. (/) <-> E. y ( x e. y /\ y e. (/) ) ) |
|
| 5 | 3 4 | mtbir | |- -. x e. U. (/) |
| 6 | 5 | nel0 | |- U. (/) = (/) |