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Theorem brcosscnvcoss 35153
 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 1822 . . 3 (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴))
21a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
3 brcoss 35150 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
4 brcoss 35150 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
54ancoms 451 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
62, 3, 53bitr4d 303 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∃wex 1742   ∈ wcel 2050   class class class wbr 4925   ≀ ccoss 34926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pr 5182 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-coss 35133 This theorem is referenced by:  cocossss  35155  cnvcosseq  35156  rncossdmcoss  35169  symrelcoss3  35179  eleccossin  35197
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