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Theorem rnsnn0 5155
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 5154 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 dm0rn0 4911 . . 3  |-  ( dom 
{ A }  =  (/)  <->  ran 
{ A }  =  (/) )
32necon3bii 2491 . 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 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   {csn 3653    X. cxp 4703   dom cdm 4705   ran crn 4706
This theorem is referenced by:  2nd2val  6162  2ndnpr  23261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716
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