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Theorem opid 4786
 Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Inference form of opidg 4785. (Contributed by FL, 28-Dec-2011.) (Proof shortened by AV, 16-Feb-2022.) (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 opid.1 . 2 𝐴 ∈ V
2 opidg 4785 . 2 (𝐴 ∈ V → ⟨𝐴, 𝐴⟩ = {{𝐴}})
31, 2ax-mp 5 1 𝐴, 𝐴⟩ = {{𝐴}}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532 This theorem is referenced by:  dmsnsnsn  6045  funopg  6359  vtxval3sn  26846  iedgval3sn  26847
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