Description: A class is not an element of and equal to a class at the same time. Variant of elneq analogously to elnotel and en2lp . (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | nelaneq | |- -. ( A e. B /\ A = B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elneq | |- ( A e. B -> A =/= B ) |
|
| 2 | orc | |- ( -. A e. B -> ( -. A e. B \/ -. A = B ) ) |
|
| 3 | neneq | |- ( A =/= B -> -. A = B ) |
|
| 4 | 3 | olcd | |- ( A =/= B -> ( -. A e. B \/ -. A = B ) ) |
| 5 | 2 4 | ja | |- ( ( A e. B -> A =/= B ) -> ( -. A e. B \/ -. A = B ) ) |
| 6 | 1 5 | ax-mp | |- ( -. A e. B \/ -. A = B ) |
| 7 | ianor | |- ( -. ( A e. B /\ A = B ) <-> ( -. A e. B \/ -. A = B ) ) |
|
| 8 | 6 7 | mpbir | |- -. ( A e. B /\ A = B ) |