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Theorem brcosscnvcoss 35559
Description: For sets, the 𝐴 and 𝐵 cosets by 𝑅 binary relation and the 𝐵 and 𝐴 cosets by 𝑅 binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
brcosscnvcoss ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem brcosscnvcoss
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 exancom 1852 . . 3 (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴))
21a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
3 brcoss 35556 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
4 brcoss 35556 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
54ancoms 459 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
62, 3, 53bitr4d 312 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wex 1771  wcel 2105   class class class wbr 5057  ccoss 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-coss 35539
This theorem is referenced by:  cocossss  35561  cnvcosseq  35562  rncossdmcoss  35575  symrelcoss3  35585  eleccossin  35603
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