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)
| 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 | 4 | necon3bi | |- ( -. { { (/) } } e. On -> On =/= _V ) |
| 6 | 1 5 | ax-mp | |- On =/= _V |