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Theorem relsnop 4949
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 4890 . 2  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
4 opex 4561 . . 3  |-  <. A ,  B >.  e.  _V
54relsn 4948 . 2  |-  ( Rel 
{ <. A ,  B >. }  <->  <. A ,  B >.  e.  ( _V  X.  _V ) )
63, 5mpbir 209 1  |-  Rel  { <. A ,  B >. }
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1756   _Vcvv 2977   {csn 3882   <.cop 3888    X. cxp 4843   Rel wrel 4850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-opab 4356  df-xp 4851  df-rel 4852
This theorem is referenced by:  cnvsn  5327  fsn  5886  imasaddfnlem  14471  ex-res  23653
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