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Theorem relsn 4946
Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
Hypothesis
Ref Expression
relsn.1  |-  A  e. 
_V
Assertion
Ref Expression
relsn  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)

Proof of Theorem relsn
StepHypRef Expression
1 df-rel 4852 . 2  |-  ( Rel 
{ A }  <->  { A }  C_  ( _V  X.  _V ) )
2 relsn.1 . . 3  |-  A  e. 
_V
32snss 3894 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  { A }  C_  ( _V  X.  _V )
)
41, 3bitr4i 244 1  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1721   _Vcvv 2924    C_ wss 3288   {csn 3782    X. cxp 4843   Rel wrel 4850
This theorem is referenced by:  relsnop  4947  relsn2  5307  setscom  13460  setsid  13471
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-v 2926  df-in 3295  df-ss 3302  df-sn 3788  df-rel 4852
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