Theorem nelsnOLD 4160

Description: Obsolete proof of nelsn 4159 as of 4-May-2021. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nelsnOLD	⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})

Proof of Theorem nelsnOLD

Step	Hyp	Ref	Expression
1		neneq 2788	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ V) → ¬ 𝐴 = 𝐵)
3		elsng 4139	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
4	3	adantl 481	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ V) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	2, 4	mtbird 314	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ V) → ¬ 𝐴 ∈ {𝐵})
6		prcnel 3191	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ {𝐵})
7	6	adantl 481	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 ∈ {𝐵})
8	5, 7	pm2.61dan 828	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-sn 4126

This theorem is referenced by: (None)