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Theorem relsn 4614
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 4516 . 2  |-  ( Rel 
{ A }  <->  { A }  C_  ( _V  X.  _V ) )
2 relsn.1 . . 3  |-  A  e. 
_V
32snss 3619 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  { A }  C_  ( _V  X.  _V )
)
41, 3bitr4i 186 1  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1465   _Vcvv 2660    C_ wss 3041   {csn 3497    X. cxp 4507   Rel wrel 4514
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-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-sn 3503  df-rel 4516
This theorem is referenced by:  relsnop  4615  relsn2m  4979
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