Theorem nelaneq 9626

Description: A class is not an element of and equal to a class at the same time. Variant of elneq 9625 analogously to elnotel 9641 and en2lp 9637. (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 9624	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq2 2853	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵))
3	2	biimparc 485	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐴)
4	1, 3	mto 200	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-reg 9616

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839

This theorem is used by:  epinid0  9629