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

Theorem rnsnm 4975
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 4974 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
dom  { A } )
2 dmmrnm 4728 . 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 1453    e. wcel 1465   _Vcvv 2660   {csn 3497    X. cxp 4507   dom cdm 4509   ran crn 4510
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator