Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  relnonrel Structured version   Visualization version   GIF version

Theorem relnonrel 43017
Description: The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
relnonrel (Rel 𝐴 ↔ (𝐴𝐴) = ∅)

Proof of Theorem relnonrel
StepHypRef Expression
1 dfrel2 6193 . . 3 (Rel 𝐴𝐴 = 𝐴)
2 eqss 3995 . . 3 (𝐴 = 𝐴 ↔ (𝐴𝐴𝐴𝐴))
31, 2bitri 275 . 2 (Rel 𝐴 ↔ (𝐴𝐴𝐴𝐴))
4 cnvcnvss 6198 . . 3 𝐴𝐴
54biantrur 530 . 2 (𝐴𝐴 ↔ (𝐴𝐴𝐴𝐴))
6 ssdif0 4364 . 2 (𝐴𝐴 ↔ (𝐴𝐴) = ∅)
73, 5, 63bitr2i 299 1 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  cdif 3944  wss 3947  c0 4323  ccnv 5677  Rel wrel 5683
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 5299  ax-nul 5306  ax-pr 5429
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-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686
This theorem is referenced by:  cnvnonrel  43018
  Copyright terms: Public domain W3C validator