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Theorem opid 4359
 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 4138 . . 3 {𝐴} = {𝐴, 𝐴}
21preq2i 4216 . 2 {{𝐴}, {𝐴}} = {{𝐴}, {𝐴, 𝐴}}
3 dfsn2 4138 . 2 {{𝐴}} = {{𝐴}, {𝐴}}
4 opid.1 . . 3 𝐴 ∈ V
54, 4dfop 4339 . 2 𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
62, 3, 53eqtr4ri 2643 1 𝐴, 𝐴⟩ = {{𝐴}}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  {cpr 4127  ⟨cop 4131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132 This theorem is referenced by:  dmsnsnsn  5531  funopg  5836  vtxval3sn  25718  iedgval3sn  25719
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