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Theorem relrn0 5112
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 5072 . 2  |-  ( Rel 
A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
2 dm0rn0 5071 . 2  |-  ( dom 
A  =  (/)  <->  ran  A  =  (/) )
31, 2syl6bb 264 1  |-  ( Rel 
A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   (/)c0 3767   dom cdm 4854   ran crn 4855   Rel wrel 4859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-dm 4864  df-rn 4865
This theorem is referenced by:  cnvsn0  5324  coeq0  5364  foconst  5821  fconst5  6137  0eusgraiff0rgracl  25514  heicant  31679
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