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Theorem opidg 4786
 Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4787. (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 4763 . . 3 ((𝐴𝑉𝐴𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
21anidms 570 . 2 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3 dfsn2 4539 . . . . 5 {𝐴} = {𝐴, 𝐴}
43eqcomi 2768 . . . 4 {𝐴, 𝐴} = {𝐴}
54preq2i 4634 . . 3 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6 dfsn2 4539 . . 3 {{𝐴}} = {{𝐴}, {𝐴}}
75, 6eqtr4i 2785 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
82, 7eqtrdi 2810 1 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  {csn 4526  {cpr 4528  ⟨cop 4532 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-dif 3864  df-un 3866  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533 This theorem is referenced by:  opid  4787  brin3  36134
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