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Theorem opid 4693
 Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Inference form of opidg 4692. (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 4692 . 2 (𝐴 ∈ V → ⟨𝐴, 𝐴⟩ = {{𝐴}})
31, 2ax-mp 5 1 𝐴, 𝐴⟩ = {{𝐴}}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1508   ∈ wcel 2051  Vcvv 3408  {csn 4435  ⟨cop 4441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442 This theorem is referenced by:  dmsnsnsn  5913  funopg  6219  vtxval3sn  26546  iedgval3sn  26547
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