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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 (𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})

Proof of Theorem nelsnOLD
StepHypRef Expression
1 neneq 2788 . . . 4 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
21adantr 480 . . 3 ((𝐴𝐵𝐴 ∈ V) → ¬ 𝐴 = 𝐵)
3 elsng 4139 . . . 4 (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
43adantl 481 . . 3 ((𝐴𝐵𝐴 ∈ V) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
52, 4mtbird 314 . 2 ((𝐴𝐵𝐴 ∈ V) → ¬ 𝐴 ∈ {𝐵})
6 prcnel 3191 . . 3 𝐴 ∈ V → ¬ 𝐴 ∈ {𝐵})
76adantl 481 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 ∈ {𝐵})
85, 7pm2.61dan 828 1 (𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})
 Colors of variables: wff setvar class 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)
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