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Theorem brnonrel 40690
 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 5084 . 2 ¬ 𝑌𝑋
2 brcnvg 5724 . . . 4 ((𝑌𝑉𝑋𝑈) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
32ancoms 462 . . 3 ((𝑋𝑈𝑌𝑉) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
4 cnvnonrel 40689 . . . 4 (𝐴𝐴) = ∅
54breqi 5041 . . 3 (𝑌(𝐴𝐴)𝑋𝑌𝑋)
63, 5bitr3di 289 . 2 ((𝑋𝑈𝑌𝑉) → (𝑋(𝐴𝐴)𝑌𝑌𝑋))
71, 6mtbiri 330 1 ((𝑋𝑈𝑌𝑉) → ¬ 𝑋(𝐴𝐴)𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ∖ cdif 3857  ∅c0 4227   class class class wbr 5035  ◡ccnv 5526 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 2113  ax-9 2121  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-cnv 5535 This theorem is referenced by: (None)
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