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Theorem brvbrvvdif 35678
 Description: Binary relation with the complement under the universal class of ordered pairs is the same as with universal complement. (Contributed by Peter Mazsa, 28-Nov-2018.)
Assertion
Ref Expression
brvbrvvdif ((𝐴𝑉𝐵𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵𝐴(V ∖ 𝑅)𝐵))

Proof of Theorem brvbrvvdif
StepHypRef Expression
1 brvvdif 35677 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵))
2 brvdif 35675 . 2 (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)
31, 2syl6bbr 292 1 ((𝐴𝑉𝐵𝑊) → (𝐴((V × V) ∖ 𝑅)𝐵𝐴(V ∖ 𝑅)𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  Vcvv 3444   ∖ cdif 3881   class class class wbr 5033   × cxp 5521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529 This theorem is referenced by: (None)
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