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Theorem relsnopg 4490
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 4435 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
2 opexg 4011 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e. 
_V )
3 relsng 4489 . . 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 165 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Rel  { <. A ,  B >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   _Vcvv 2610   {csn 3416   <.cop 3419    X. cxp 4389   Rel wrel 4396
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860  df-xp 4397  df-rel 4398
This theorem is referenced by: (None)
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