ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rnsnm Unicode version

Theorem rnsnm 5009
Description: The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
Assertion
Ref Expression
rnsnm  |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
ran  { A } )
Distinct variable group:    x, A

Proof of Theorem rnsnm
StepHypRef Expression
1 dmsnm 5008 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
dom  { A } )
2 dmmrnm 4762 . 2  |-  ( E. x  x  e.  dom  { A }  <->  E. x  x  e.  ran  { A } )
31, 2bitri 183 1  |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
ran  { A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1469    e. wcel 1481   _Vcvv 2687   {csn 3528    X. cxp 4541   dom cdm 4543   ran crn 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-xp 4549  df-cnv 4551  df-dm 4553  df-rn 4554
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator