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Theorem nelbr 47286
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 2831 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
21notbid 318 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝑥𝑦 ↔ ¬ 𝐴𝐵))
3 df-nelbr 47284 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
42, 3brabga 5539 1 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143   _∉ cnelbr 47283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-nelbr 47284
This theorem is referenced by:  nelbrim  47287  nelbrnel  47288
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