Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  symrelcoss3 Structured version   Visualization version   GIF version

Theorem symrelcoss3 36973
Description: The class of cosets by 𝑅 is symmetric, see dfsymrel3 37058. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
Assertion
Ref Expression
symrelcoss3 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)

Proof of Theorem symrelcoss3
StepHypRef Expression
1 brcosscnvcoss 36942 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦𝑦𝑅𝑥))
21el2v 3452 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
32biimpi 215 . . 3 (𝑥𝑅𝑦𝑦𝑅𝑥)
43gen2 1799 . 2 𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)
5 relcoss 36931 . 2 Rel ≀ 𝑅
64, 5pm3.2i 472 1 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  Vcvv 3444   class class class wbr 5106  Rel wrel 5639  ccoss 36680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-coss 36919
This theorem is referenced by:  symrelcoss2  36974  eqvrelcoss3  37126
  Copyright terms: Public domain W3C validator