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Theorem relsn 5014
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 4920 . 2  |-  ( Rel 
{ A }  <->  { A }  C_  ( _V  X.  _V ) )
2 relsn.1 . . 3  |-  A  e. 
_V
32snss 3955 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  { A }  C_  ( _V  X.  _V )
)
41, 3bitr4i 245 1  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1728   _Vcvv 2965    C_ wss 3309   {csn 3843    X. cxp 4911   Rel wrel 4918
This theorem is referenced by:  relsnop  5015  relsn2  5375  setscom  13535  setsid  13546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-v 2967  df-in 3316  df-ss 3323  df-sn 3849  df-rel 4920
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