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Theorem opid 4804
Description: The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Inference form of opidg 4803. (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 4803 . 2 (𝐴 ∈ V → ⟨𝐴, 𝐴⟩ = {{𝐴}})
31, 2ax-mp 5 1 𝐴, 𝐴⟩ = {{𝐴}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2110  Vcvv 3408  {csn 4541  cop 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548
This theorem is referenced by:  dmsnsnsn  6083  funopg  6414  vtxval3sn  27134  iedgval3sn  27135
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