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Theorem brvdif 35054
 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 5256 . 2 𝐴V𝐵
2 brdif 5015 . 2 (𝐴(V ∖ 𝑅)𝐵 ↔ (𝐴V𝐵 ∧ ¬ 𝐴𝑅𝐵))
31, 2mpbiran 705 1 (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 207  Vcvv 3437   ∖ cdif 3856   class class class wbr 4962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963 This theorem is referenced by:  brvdif2  35055  brvbrvvdif  35057  dfssr2  35270
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