Theorem nelsnOLD 4213

Description: Obsolete proof of nelsn 4212 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 2800	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
2	1	adantr 481	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ V) → ¬ 𝐴 = 𝐵)
3		elsng 4191	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
4	3	adantl 482	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ V) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	2, 4	mtbird 315	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ V) → ¬ 𝐴 ∈ {𝐵})
6		prcnel 3218	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ {𝐵})
7	6	adantl 482	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 ∈ {𝐵})
8	5, 7	pm2.61dan 832	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-sn 4178

This theorem is referenced by: (None)