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Theorem cnvnonrel 43578
Description: The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)
Assertion
Ref Expression
cnvnonrel (𝐴𝐴) = ∅

Proof of Theorem cnvnonrel
StepHypRef Expression
1 cnvdif 6166 . 2 (𝐴𝐴) = (𝐴𝐴)
2 relcnv 6125 . . 3 Rel 𝐴
3 relnonrel 43577 . . 3 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
42, 3mpbi 230 . 2 (𝐴𝐴) = ∅
51, 4eqtri 2763 1 (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cdif 3960  c0 4339  ccnv 5688  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by:  brnonrel  43579  dmnonrel  43580  resnonrel  43582  cononrel1  43584  cononrel2  43585  clcnvlem  43613  cnvrcl0  43615
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