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Theorem opid 3771
Description: The ordered pair  <. A ,  A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
Hypothesis
Ref Expression
opid.1  |-  A  e. 
_V
Assertion
Ref Expression
opid  |-  <. A ,  A >.  =  { { A } }

Proof of Theorem opid
StepHypRef Expression
1 dfsn2 3585 . . . 4  |-  { A }  =  { A ,  A }
21eqcomi 2168 . . 3  |-  { A ,  A }  =  { A }
32preq2i 3652 . 2  |-  { { A } ,  { A ,  A } }  =  { { A } ,  { A } }
4 opid.1 . . 3  |-  A  e. 
_V
54, 4dfop 3752 . 2  |-  <. A ,  A >.  =  { { A } ,  { A ,  A } }
6 dfsn2 3585 . 2  |-  { { A } }  =  { { A } ,  { A } }
73, 5, 63eqtr4i 2195 1  |-  <. A ,  A >.  =  { { A } }
Colors of variables: wff set class
Syntax hints:    = wceq 1342    e. wcel 2135   _Vcvv 2722   {csn 3571   {cpr 3572   <.cop 3574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-un 3116  df-sn 3577  df-pr 3578  df-op 3580
This theorem is referenced by:  dmsnsnsng  5076  funopg  5217
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