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Theorem nelbr 41811
 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 2840 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
21notbid 307 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝑥𝑦 ↔ ¬ 𝐴𝐵))
3 df-nelbr 41809 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
42, 3brabga 5123 1 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145   class class class wbr 4787   _∉ cnelbr 41808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-nelbr 41809 This theorem is referenced by:  nelbrim  41812  nelbrnel  41813
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