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Theorem cnvnonrel 38420
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 5679 . 2 (𝐴𝐴) = (𝐴𝐴)
2 relcnv 5643 . . 3 Rel 𝐴
3 relnonrel 38419 . . 3 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
42, 3mpbi 220 . 2 (𝐴𝐴) = ∅
51, 4eqtri 2793 1 (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cdif 3720  c0 4063  ccnv 5249  Rel wrel 5255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258
This theorem is referenced by:  brnonrel  38421  dmnonrel  38422  resnonrel  38424  cononrel1  38426  cononrel2  38427  clcnvlem  38456  cnvrcl0  38458
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