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Theorem opid 4559
Description: The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Inference form of opidg 4558. (Contributed by FL, 28-Dec-2011.) (Prove 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 4558 . 2 (𝐴 ∈ V → ⟨𝐴, 𝐴⟩ = {{𝐴}})
31, 2ax-mp 5 1 𝐴, 𝐴⟩ = {{𝐴}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4316  cop 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323
This theorem is referenced by:  dmsnsnsn  5755  funopg  6065  vtxval3sn  26156  iedgval3sn  26157
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