Theorem nelaneqOLD 9515

Description: Obsolete version of nelaneq 9514 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 9512	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq2 2829	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵))
3	1, 2	mtbii 327	. . 3 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵)
4	3	con2i 139	. 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵)
5		imnan 400	. 2 ⊢ ((𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵))
6	4, 5	mpbi 231	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-reg 9504

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815

This theorem is referenced by: (None)