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Theorem relsn2m 4821
Description: A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)
Hypothesis
Ref Expression
relsn2m.1  |-  A  e. 
_V
Assertion
Ref Expression
relsn2m  |-  ( Rel 
{ A }  <->  E. x  x  e.  dom  { A } )
Distinct variable group:    x, A

Proof of Theorem relsn2m
StepHypRef Expression
1 relsn2m.1 . . 3  |-  A  e. 
_V
21relsn 4471 . 2  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)
3 dmsnm 4816 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
dom  { A } )
42, 3bitri 182 1  |-  ( Rel 
{ A }  <->  E. x  x  e.  dom  { A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1422    e. wcel 1434   _Vcvv 2602   {csn 3406    X. cxp 4369   dom cdm 4371   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-dm 4381
This theorem is referenced by: (None)
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