Theorem nelaneqOLD 9553

Description: Obsolete version of nelaneq 9552 as of 22-Apr-2026. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) (Proof shortened by TM, 31-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem nelaneqOLD

Step	Hyp	Ref	Expression
1		elirr 9550	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq2 2854	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵))
3	1, 2	mtbii 329	. . 3 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵)
4	3	con2i 140	. 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵)
5		imnan 404	. 2 ⊢ ((𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵))
6	4, 5	mpbi 233	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-reg 9542

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840

This theorem is referenced by: (None)