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Theorem relsn 4471
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 4380 . 2  |-  ( Rel 
{ A }  <->  { A }  C_  ( _V  X.  _V ) )
2 relsn.1 . . 3  |-  A  e. 
_V
32snss 3522 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  { A }  C_  ( _V  X.  _V )
)
41, 3bitr4i 180 1  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 102    e. wcel 1409   _Vcvv 2574    C_ wss 2945   {csn 3403    X. cxp 4371   Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-sn 3409  df-rel 4380
This theorem is referenced by:  relsnop  4472  relsn2m  4819
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