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Theorem dmnonrel 39828
Description: The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
dmnonrel dom (𝐴𝐴) = ∅

Proof of Theorem dmnonrel
StepHypRef Expression
1 dfdm4 5757 . 2 dom (𝐴𝐴) = ran (𝐴𝐴)
2 cnvnonrel 39826 . . 3 (𝐴𝐴) = ∅
32rneqi 5800 . 2 ran (𝐴𝐴) = ran ∅
4 rn0 5789 . 2 ran ∅ = ∅
51, 3, 43eqtri 2845 1 dom (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cdif 3930  c0 4288  ccnv 5547  dom cdm 5548  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  rnnonrel  39829
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