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Theorem cnvnonrel 43601
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 6163 . 2 (𝐴𝐴) = (𝐴𝐴)
2 relcnv 6122 . . 3 Rel 𝐴
3 relnonrel 43600 . . 3 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
42, 3mpbi 230 . 2 (𝐴𝐴) = ∅
51, 4eqtri 2765 1 (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cdif 3948  c0 4333  ccnv 5684  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  brnonrel  43602  dmnonrel  43603  resnonrel  43605  cononrel1  43607  cononrel2  43608  clcnvlem  43636  cnvrcl0  43638
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