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Theorem opid 4389
Description: The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) (Avoid depending on this detail.)
Hypothesis
Ref Expression
opid.1 𝐴 ∈ V
Assertion
Ref Expression
opid 𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid
StepHypRef Expression
1 dfsn2 4161 . . 3 {𝐴} = {𝐴, 𝐴}
21preq2i 4242 . 2 {{𝐴}, {𝐴}} = {{𝐴}, {𝐴, 𝐴}}
3 dfsn2 4161 . 2 {{𝐴}} = {{𝐴}, {𝐴}}
4 opid.1 . . 3 𝐴 ∈ V
54, 4dfop 4369 . 2 𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
62, 3, 53eqtr4ri 2654 1 𝐴, 𝐴⟩ = {{𝐴}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148  {cpr 4150  cop 4154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155
This theorem is referenced by:  dmsnsnsn  5572  funopg  5880  vtxval3sn  25835  iedgval3sn  25836
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