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Theorem relsnop 4947
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 4891 . 2  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
4 opex 4395 . . 3  |-  <. A ,  B >.  e.  _V
54relsn 4946 . 2  |-  ( Rel 
{ <. A ,  B >. }  <->  <. A ,  B >.  e.  ( _V  X.  _V ) )
63, 5mpbir 201 1  |-  Rel  { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   _Vcvv 2924   {csn 3782   <.cop 3785    X. cxp 4843   Rel wrel 4850
This theorem is referenced by:  cnvsn  5319  fsn  5873  imasaddfnlem  13716  ex-res  21710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-opab 4235  df-xp 4851  df-rel 4852
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