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Theorem relsnop 4791
Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
relsn.1  |-  A  e. 
_V
relsnop.2  |-  B  e. 
_V
Assertion
Ref Expression
relsnop  |-  Rel  { <. A ,  B >. }

Proof of Theorem relsnop
StepHypRef Expression
1 relsn.1 . . 3  |-  A  e. 
_V
2 relsnop.2 . . 3  |-  B  e. 
_V
31, 2opelvv 4735 . 2  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
4 opex 4237 . . 3  |-  <. A ,  B >.  e.  _V
54relsn 4790 . 2  |-  ( Rel 
{ <. A ,  B >. }  <->  <. A ,  B >.  e.  ( _V  X.  _V ) )
63, 5mpbir 200 1  |-  Rel  { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643    X. cxp 4687   Rel wrel 4694
This theorem is referenced by:  cnvsn  5155  fsn  5696  imasaddfnlem  13430  ex-res  20828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696
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