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Theorem relsn2 5280
Description: A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.)
Hypothesis
Ref Expression
relsn2.1  |-  A  e. 
_V
Assertion
Ref Expression
relsn2  |-  ( Rel 
{ A }  <->  dom  { A }  =/=  (/) )

Proof of Theorem relsn2
StepHypRef Expression
1 relsn2.1 . . 3  |-  A  e. 
_V
21relsn 4919 . 2  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)
3 dmsnn0 5275 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
42, 3bitri 241 1  |-  ( Rel 
{ A }  <->  dom  { A }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1717    =/= wne 2550   _Vcvv 2899   (/)c0 3571   {csn 3757    X. cxp 4816   dom cdm 4818   Rel wrel 4823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-dm 4828
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