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Theorem brvdif 37633
Description: Binary relation with universal complement is the negation of the relation. (Contributed by Peter Mazsa, 1-Jul-2018.)
Assertion
Ref Expression
brvdif (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)

Proof of Theorem brvdif
StepHypRef Expression
1 brv 5463 . 2 𝐴V𝐵
2 brdif 5192 . 2 (𝐴(V ∖ 𝑅)𝐵 ↔ (𝐴V𝐵 ∧ ¬ 𝐴𝑅𝐵))
31, 2mpbiran 706 1 (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  Vcvv 3466  cdif 3938   class class class wbr 5139
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 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
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 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140
This theorem is referenced by:  brvdif2  37634  brvbrvvdif  37636  dfssr2  37873
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