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

Theorem nelbrnelim 46651
Description: If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
Assertion
Ref Expression
nelbrnelim (𝐴 _∉ 𝐵𝐴𝐵)

Proof of Theorem nelbrnelim
StepHypRef Expression
1 nelbrim 46649 . 2 (𝐴 _∉ 𝐵 → ¬ 𝐴𝐵)
2 df-nel 3043 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
31, 2sylibr 233 1 (𝐴 _∉ 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2099  wnel 3042   class class class wbr 5142   _∉ cnelbr 46645
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nel 3043  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-nelbr 46646
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator