Description: The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007) (Proof shortened by Mario Carneiro, 10-Jan-2013) (Proof shortened by Wolf Lammen, 27-May-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | onnev | |- On =/= _V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snsn0non | |- -. { { (/) } } e. On |
|
| 2 | snex | |- { { (/) } } e. _V |
|
| 3 | id | |- ( On = _V -> On = _V ) |
|
| 4 | 2 3 | eleqtrrid | |- ( On = _V -> { { (/) } } e. On ) |
| 5 | 1 4 | mto | |- -. On = _V |
| 6 | 5 | neir | |- On =/= _V |