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Theorem brnonrel 37373
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 4661 . 2 ¬ 𝑌𝑋
2 cnvnonrel 37372 . . . 4 (𝐴𝐴) = ∅
32breqi 4619 . . 3 (𝑌(𝐴𝐴)𝑋𝑌𝑋)
4 brcnvg 5263 . . . 4 ((𝑌𝑉𝑋𝑈) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
54ancoms 469 . . 3 ((𝑋𝑈𝑌𝑉) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
63, 5syl5rbbr 275 . 2 ((𝑋𝑈𝑌𝑉) → (𝑋(𝐴𝐴)𝑌𝑌𝑋))
71, 6mtbiri 317 1 ((𝑋𝑈𝑌𝑉) → ¬ 𝑋(𝐴𝐴)𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1987  cdif 3552  c0 3891   class class class wbr 4613  ccnv 5073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082
This theorem is referenced by: (None)
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