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Theorem opidg 4897
Description: The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Closed form of opid 4898. (Contributed by Peter Mazsa, 22-Jul-2019.) (Avoid depending on this detail.)
Assertion
Ref Expression
opidg (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Proof of Theorem opidg
StepHypRef Expression
1 dfopg 4876 . . 3 ((𝐴𝑉𝐴𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
21anidms 565 . 2 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3 dfsn2 4645 . . . . 5 {𝐴} = {𝐴, 𝐴}
43eqcomi 2737 . . . 4 {𝐴, 𝐴} = {𝐴}
54preq2i 4746 . . 3 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6 dfsn2 4645 . . 3 {{𝐴}} = {{𝐴}, {𝐴}}
75, 6eqtr4i 2759 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
82, 7eqtrdi 2784 1 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {csn 4632  {cpr 4634  cop 4638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639
This theorem is referenced by:  opid  4898  brin3  37914
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