Theorem nelaneq 9516

Description: A class is not an element of and equal to a class at the same time. Variant of elneq 9515 analogously to elnotel 9531 and en2lp 9527. (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 9514	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq2 2825	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵))
3	2	biimparc 479	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐴)
4	1, 3	mto 197	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-reg 9507

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811

This theorem is referenced by:  epinid0  9519