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Theorem relsng 4529
Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
Assertion
Ref Expression
relsng  |-  ( A  e.  V  ->  ( Rel  { A }  <->  A  e.  ( _V  X.  _V )
) )

Proof of Theorem relsng
StepHypRef Expression
1 snssg 3568 . 2  |-  ( A  e.  V  ->  ( A  e.  ( _V  X.  _V )  <->  { A }  C_  ( _V  X.  _V ) ) )
2 df-rel 4435 . 2  |-  ( Rel 
{ A }  <->  { A }  C_  ( _V  X.  _V ) )
31, 2syl6rbbr 197 1  |-  ( A  e.  V  ->  ( Rel  { A }  <->  A  e.  ( _V  X.  _V )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1438   _Vcvv 2619    C_ wss 2997   {csn 3441    X. cxp 4426   Rel wrel 4433
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-sn 3447  df-rel 4435
This theorem is referenced by:  relsnopg  4530
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