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Theorem cnvnonrel 40459
 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 5973 . 2 (𝐴𝐴) = (𝐴𝐴)
2 relcnv 5938 . . 3 Rel 𝐴
3 relnonrel 40458 . . 3 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
42, 3mpbi 233 . 2 (𝐴𝐴) = ∅
51, 4eqtri 2821 1 (𝐴𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∖ cdif 3880  ∅c0 4246  ◡ccnv 5522  Rel wrel 5528 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-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-cnv 5531 This theorem is referenced by:  brnonrel  40460  dmnonrel  40461  resnonrel  40463  cononrel1  40465  cononrel2  40466  clcnvlem  40494  cnvrcl0  40496
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