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Theorem relrn0 5120
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 5079 . 2  |-  ( Rel 
A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
2 dm0rn0 5078 . 2  |-  ( dom 
A  =  (/)  <->  ran  A  =  (/) )
31, 2syl6bb 253 1  |-  ( Rel 
A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   (/)c0 3620   dom cdm 4870   ran crn 4871   Rel wrel 4875
This theorem is referenced by:  cnvsn0  5330  foconst  5656  fconst5  5941  coeq0  26791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881
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