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Theorem dmnonrel 43051
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 5902 . 2 dom (𝐴𝐴) = ran (𝐴𝐴)
2 cnvnonrel 43049 . . 3 (𝐴𝐴) = ∅
32rneqi 5943 . 2 ran (𝐴𝐴) = ran ∅
4 rn0 5932 . 2 ran ∅ = ∅
51, 3, 43eqtri 2760 1 dom (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cdif 3946  c0 4326  ccnv 5681  dom cdm 5682  ran crn 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693
This theorem is referenced by:  rnnonrel  43052
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