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Theorem relrn0 4695
Description: A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
relrn0 (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Proof of Theorem relrn0
StepHypRef Expression
1 reldm0 4654 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2 dm0rn0 4653 . 2 (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
31, 2syl6bb 194 1 (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  c0 3286  dom cdm 4438  ran crn 4439  Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  cnvsn0  4899
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