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Theorem relcoss 35660
Description: Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
relcoss Rel ≀ 𝑅

Proof of Theorem relcoss
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-coss 35651 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
21relopabi 5687 1 Rel ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 398  wex 1774   class class class wbr 5057  Rel wrel 5553  ccoss 35445
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-opab 5120  df-xp 5554  df-rel 5555  df-coss 35651
This theorem is referenced by:  relcoels  35661  cocossss  35673  cnvcosseq  35674  refrelcoss3  35695  symrelcoss3  35697  1cosscnvxrn  35707  eleccossin  35715  cosselrels  35728  cnvrefrelcoss2  35765  eqvrelcoss3  35845
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