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Theorem rnsnn0 5328
Description: The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
rnsnn0  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )

Proof of Theorem rnsnn0
StepHypRef Expression
1 dmsnn0 5327 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 dm0rn0 5078 . . 3  |-  ( dom 
{ A }  =  (/)  <->  ran 
{ A }  =  (/) )
32necon3bii 2630 . 2  |-  ( dom 
{ A }  =/=  (/)  <->  ran 
{ A }  =/=  (/) )
41, 3bitri 241 1  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620   {csn 3806    X. cxp 4868   dom cdm 4870   ran crn 4871
This theorem is referenced by:  2nd2val  6365  2ndnpr  24086
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-cnv 4878  df-dm 4880  df-rn 4881
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