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Theorem relsnop 4532
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 4476 . 2  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
41, 2opex 4047 . . 3  |-  <. A ,  B >.  e.  _V
54relsn 4531 . 2  |-  ( Rel 
{ <. A ,  B >. }  <->  <. A ,  B >.  e.  ( _V  X.  _V ) )
63, 5mpbir 144 1  |-  Rel  { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:    e. wcel 1438   _Vcvv 2619   {csn 3441   <.cop 3444    X. cxp 4426   Rel wrel 4433
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434  df-rel 4435
This theorem is referenced by:  cnvsn  4900  fsn  5453
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