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Theorem nelbr 45754
Description: The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)
Assertion
Ref Expression
nelbr ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))

Proof of Theorem nelbr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq12 2822 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
21notbid 317 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝑥𝑦 ↔ ¬ 𝐴𝐵))
3 df-nelbr 45752 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
42, 3brabga 5527 1 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106   class class class wbr 5141   _∉ cnelbr 45751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-nelbr 45752
This theorem is referenced by:  nelbrim  45755  nelbrnel  45756
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