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Theorem opid 3830
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 3667 . . . 4  |-  { A }  =  { A ,  A }
21eqcomi 2300 . . 3  |-  { A ,  A }  =  { A }
32preq2i 3723 . 2  |-  { { A } ,  { A ,  A } }  =  { { A } ,  { A } }
4 opid.1 . . 3  |-  A  e. 
_V
54, 4dfop 3811 . 2  |-  <. A ,  A >.  =  { { A } ,  { A ,  A } }
6 dfsn2 3667 . 2  |-  { { A } }  =  { { A } ,  { A } }
73, 5, 63eqtr4i 2326 1  |-  <. A ,  A >.  =  { { A } }
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   {cpr 3654   <.cop 3656
This theorem is referenced by:  dmsnsnsn  5167  funopg  5302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662
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