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Theorem opid 3595
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 3417 . . . 4  |-  { A }  =  { A ,  A }
21eqcomi 2060 . . 3  |-  { A ,  A }  =  { A }
32preq2i 3479 . 2  |-  { { A } ,  { A ,  A } }  =  { { A } ,  { A } }
4 opid.1 . . 3  |-  A  e. 
_V
54, 4dfop 3576 . 2  |-  <. A ,  A >.  =  { { A } ,  { A ,  A } }
6 dfsn2 3417 . 2  |-  { { A } }  =  { { A } ,  { A } }
73, 5, 63eqtr4i 2086 1  |-  <. A ,  A >.  =  { { A } }
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409   _Vcvv 2574   {csn 3403   {cpr 3404   <.cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412
This theorem is referenced by:  dmsnsnsng  4826  funopg  4962
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