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Theorem symrelcoss2 37849
Description: The class of cosets by 𝑅 is symmetric, see dfsymrel2 37932. (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
symrelcoss2 (𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Proof of Theorem symrelcoss2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symrelcoss3 37848 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)
2 cnvsym 6107 . . 3 (𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32anbi1i 623 . 2 ((𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅))
41, 3mpbir 230 1 (𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531  wss 3943   class class class wbr 5141  ccnv 5668  Rel wrel 5674  ccoss 37556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-coss 37794
This theorem is referenced by:  symrelcoss  37943
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