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Theorem brcosscnvcoss 36553
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 1868 . . 3 (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴))
21a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
3 brcoss 36550 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
4 brcoss 36550 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
54ancoms 459 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
62, 3, 53bitr4d 311 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wex 1786  wcel 2110   class class class wbr 5079  ccoss 36329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-coss 36533
This theorem is referenced by:  cocossss  36555  cnvcosseq  36556  rncossdmcoss  36569  symrelcoss3  36579  eleccossin  36597
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