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Theorem relrn0 5049
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 5009 . 2  |-  ( Rel 
A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
2 dm0rn0 5008 . 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 3699   dom cdm 4791   ran crn 4792   Rel wrel 4796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pr 4598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-rab 2718  df-v 3019  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-br 4362  df-opab 4421  df-xp 4797  df-rel 4798  df-cnv 4799  df-dm 4801  df-rn 4802
This theorem is referenced by:  cnvsn0  5261  coeq0  5301  foconst  5759  fconst5  6076  0eusgraiff0rgracl  25606  heicant  31882  uhgriedg0edg0  39058  edg0usgr  39068  usgr1v0edg  39071
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