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Theorem opid 4004
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 3830 . . . 4  |-  { A }  =  { A ,  A }
21eqcomi 2442 . . 3  |-  { A ,  A }  =  { A }
32preq2i 3889 . 2  |-  { { A } ,  { A ,  A } }  =  { { A } ,  { A } }
4 opid.1 . . 3  |-  A  e. 
_V
54, 4dfop 3985 . 2  |-  <. A ,  A >.  =  { { A } ,  { A ,  A } }
6 dfsn2 3830 . 2  |-  { { A } }  =  { { A } ,  { A } }
73, 5, 63eqtr4i 2468 1  |-  <. A ,  A >.  =  { { A } }
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   {cpr 3817   <.cop 3819
This theorem is referenced by:  dmsnsnsn  5350  funopg  5487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825
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