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Theorem relrn0 4873
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 4829 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2 dm0rn0 4828 . 2 (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
31, 2bitrdi 195 1 (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348  c0 3414  dom cdm 4611  ran crn 4612  Rel wrel 4616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622
This theorem is referenced by:  cnvsn0  5079
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