Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nelbr Structured version   Visualization version   GIF version

Theorem nelbr 47181
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 2827 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
21notbid 318 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝑥𝑦 ↔ ¬ 𝐴𝐵))
3 df-nelbr 47179 . 2 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
42, 3brabga 5537 1 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104   class class class wbr 5150   _∉ cnelbr 47178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5151  df-opab 5213  df-nelbr 47179
This theorem is referenced by:  nelbrim  47182  nelbrnel  47183
  Copyright terms: Public domain W3C validator