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Theorem relsnopg 4763
Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.)
Assertion
Ref Expression
relsnopg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Rel  { <. A ,  B >. } )

Proof of Theorem relsnopg
StepHypRef Expression
1 opelvvg 4708 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
2 opexg 4257 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e. 
_V )
3 relsng 4762 . . 3  |-  ( <. A ,  B >.  e. 
_V  ->  ( Rel  { <. A ,  B >. }  <->  <. A ,  B >.  e.  ( _V  X.  _V ) ) )
42, 3syl 14 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( Rel  { <. A ,  B >. }  <->  <. A ,  B >.  e.  ( _V 
X.  _V ) ) )
51, 4mpbird 167 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Rel  { <. A ,  B >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2164   _Vcvv 2760   {csn 3618   <.cop 3621    X. cxp 4657   Rel wrel 4664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-xp 4665  df-rel 4666
This theorem is referenced by:  imasaddfnlemg  12897
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