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Theorem rnsnn0 5317
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 5316 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 dm0rn0 5068 . . 3  |-  ( dom 
{ A }  =  (/)  <->  ran 
{ A }  =  (/) )
32necon3bii 2652 . 2  |-  ( dom 
{ A }  =/=  (/)  <->  ran 
{ A }  =/=  (/) )
41, 3bitri 249 1  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1756    =/= wne 2618   _Vcvv 2984   (/)c0 3649   {csn 3889    X. cxp 4850   dom cdm 4852   ran crn 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-xp 4858  df-cnv 4860  df-dm 4862  df-rn 4863
This theorem is referenced by:  2ndnpr  6594  2nd2val  6615
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