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Theorem opid 3814
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 3654 . . . 4  |-  { A }  =  { A ,  A }
21eqcomi 2287 . . 3  |-  { A ,  A }  =  { A }
32preq2i 3710 . 2  |-  { { A } ,  { A ,  A } }  =  { { A } ,  { A } }
4 opid.1 . . 3  |-  A  e. 
_V
54, 4dfop 3795 . 2  |-  <. A ,  A >.  =  { { A } ,  { A ,  A } }
6 dfsn2 3654 . 2  |-  { { A } }  =  { { A } ,  { A } }
73, 5, 63eqtr4i 2313 1  |-  <. A ,  A >.  =  { { A } }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   {cpr 3641   <.cop 3643
This theorem is referenced by:  dmsnsnsn  5151  funopg  5286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
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