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Theorem cnvnonrel 37361
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 5502 . 2 (𝐴𝐴) = (𝐴𝐴)
2 relcnv 5466 . . 3 Rel 𝐴
3 relnonrel 37360 . . 3 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
42, 3mpbi 220 . 2 (𝐴𝐴) = ∅
51, 4eqtri 2648 1 (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cdif 3557  c0 3896  ccnv 5078  Rel wrel 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087
This theorem is referenced by:  brnonrel  37362  dmnonrel  37363  resnonrel  37365  cononrel1  37367  cononrel2  37368  clcnvlem  37397  cnvrcl0  37399
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