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Theorem brnonrel 38389
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 4845 . 2 ¬ 𝑌𝑋
2 cnvnonrel 38388 . . . 4 (𝐴𝐴) = ∅
32breqi 4802 . . 3 (𝑌(𝐴𝐴)𝑋𝑌𝑋)
4 brcnvg 5450 . . . 4 ((𝑌𝑉𝑋𝑈) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
54ancoms 468 . . 3 ((𝑋𝑈𝑌𝑉) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
63, 5syl5rbbr 275 . 2 ((𝑋𝑈𝑌𝑉) → (𝑋(𝐴𝐴)𝑌𝑌𝑋))
71, 6mtbiri 316 1 ((𝑋𝑈𝑌𝑉) → ¬ 𝑋(𝐴𝐴)𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wcel 2131  cdif 3704  c0 4050   class class class wbr 4796  ccnv 5257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-xp 5264  df-rel 5265  df-cnv 5266
This theorem is referenced by: (None)
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