Theorem nelaneq 9565

Description: A class is not an element of and equal to a class at the same time. Variant of elneq 9564 analogously to elnotel 9580 and en2lp 9576. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) (Proof shortened by TM, 31-Dec-2025.) (Proof shortened by SN, 22-Apr-2026.)

Assertion

Ref	Expression
nelaneq	⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

Proof of Theorem nelaneq

Step	Hyp	Ref	Expression
1		elirr 9563	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq2 2852	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵))
3	2	biimparc 484	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐴)
4	1, 3	mto 200	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-reg 9555

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838

This theorem is referenced by:  epinid0  9568