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Metamath Proof Explorer

Theorem nelaneq

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 ∈ B ∧ A = B

Proof

Step Hyp Ref Expression
1 elirr ⊢ ¬ A ∈ A
2 eleq2 ⊢ A = B → A ∈ A ↔ A ∈ B
3 1 2 mtbii ⊢ A = B → ¬ A ∈ B
4 3 con2i ⊢ A ∈ B → ¬ A = B
5 imnan ⊢ A ∈ B → ¬ A = B ↔ ¬ A ∈ B ∧ A = B
6 4 5 mpbi ⊢ ¬ A ∈ B ∧ A = B