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Theorem brnonrel 43068
Description: A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)
Assertion
Ref Expression
brnonrel ((𝑋𝑈𝑌𝑉) → ¬ 𝑋(𝐴𝐴)𝑌)

Proof of Theorem brnonrel
StepHypRef Expression
1 br0 5201 . 2 ¬ 𝑌𝑋
2 brcnvg 5886 . . . 4 ((𝑌𝑉𝑋𝑈) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
32ancoms 457 . . 3 ((𝑋𝑈𝑌𝑉) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
4 cnvnonrel 43067 . . . 4 (𝐴𝐴) = ∅
54breqi 5158 . . 3 (𝑌(𝐴𝐴)𝑋𝑌𝑋)
63, 5bitr3di 285 . 2 ((𝑋𝑈𝑌𝑉) → (𝑋(𝐴𝐴)𝑌𝑌𝑋))
71, 6mtbiri 326 1 ((𝑋𝑈𝑌𝑉) → ¬ 𝑋(𝐴𝐴)𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2098  cdif 3946  c0 4326   class class class wbr 5152  ccnv 5681
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690
This theorem is referenced by: (None)
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