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Theorem brnonrel 36914
 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 4631 . 2 ¬ 𝑌𝑋
2 cnvnonrel 36913 . . . 4 (𝐴𝐴) = ∅
32breqi 4589 . . 3 (𝑌(𝐴𝐴)𝑋𝑌𝑋)
4 brcnvg 5225 . . . 4 ((𝑌𝑉𝑋𝑈) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
54ancoms 468 . . 3 ((𝑋𝑈𝑌𝑉) → (𝑌(𝐴𝐴)𝑋𝑋(𝐴𝐴)𝑌))
63, 5syl5rbbr 274 . 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 195   ∧ wa 383   ∈ wcel 1977   ∖ cdif 3537  ∅c0 3874   class class class wbr 4583  ◡ccnv 5037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by: (None)
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