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Theorem cnvnonrel 36913
 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 5458 . 2 (𝐴𝐴) = (𝐴𝐴)
2 relcnv 5422 . . 3 Rel 𝐴
3 relnonrel 36912 . . 3 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
42, 3mpbi 219 . 2 (𝐴𝐴) = ∅
51, 4eqtri 2632 1 (𝐴𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∖ cdif 3537  ∅c0 3874  ◡ccnv 5037  Rel wrel 5043 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:  brnonrel  36914  dmnonrel  36915  resnonrel  36917  cononrel1  36919  cononrel2  36920  clcnvlem  36949  cnvrcl0  36951
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