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Theorem rnsnn0 5480
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 5479 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 dm0rn0 5229 . . 3  |-  ( dom 
{ A }  =  (/)  <->  ran 
{ A }  =  (/) )
32necon3bii 2725 . 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 1819    =/= wne 2652   _Vcvv 3109   (/)c0 3793   {csn 4032    X. cxp 5006   dom cdm 5008   ran crn 5009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-cnv 5016  df-dm 5018  df-rn 5019
This theorem is referenced by:  2ndnpr  6804  2nd2val  6826
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