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) (Proof shortened by TM, 31-Dec-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | nelaneq | |- -. ( A e. B /\ A = B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elirr | |- -. A e. A |
|
| 2 | eleq2 | |- ( A = B -> ( A e. A <-> A e. B ) ) |
|
| 3 | 1 2 | mtbii | |- ( A = B -> -. A e. B ) |
| 4 | 3 | con2i | |- ( A e. B -> -. A = B ) |
| 5 | imnan | |- ( ( A e. B -> -. A = B ) <-> -. ( A e. B /\ A = B ) ) |
|
| 6 | 4 5 | mpbi | |- -. ( A e. B /\ A = B ) |