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 )