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Theorem nelbr 46280
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 2821 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
21notbid 317 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝑥𝑦 ↔ ¬ 𝐴𝐵))
3 df-nelbr 46278 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
42, 3brabga 5533 1 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104   class class class wbr 5147   _∉ cnelbr 46277
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-nelbr 46278
This theorem is referenced by:  nelbrim  46281  nelbrnel  46282
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