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Theorem rnsnn0 5139
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 5138 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 dm0rn0 4895 . . 3  |-  ( dom 
{ A }  =  (/)  <->  ran 
{ A }  =  (/) )
32necon3bii 2478 . 2  |-  ( dom 
{ A }  =/=  (/)  <->  ran 
{ A }  =/=  (/) )
41, 3bitri 240 1  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {csn 3640    X. cxp 4687   dom cdm 4689   ran crn 4690
This theorem is referenced by:  2nd2val  6146  2ndnpr  23246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700
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