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Theorem opidg 40624
Description: The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Closed form of opid 4396. (Contributed by AV, 18-Sep-2020.) (Revised by AV, 18-Sep-2021.)
Assertion
Ref Expression
opidg (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Proof of Theorem opidg
StepHypRef Expression
1 dfsn2 4168 . . . 4 {𝐴} = {𝐴, 𝐴}
21eqcomi 2630 . . 3 {𝐴, 𝐴} = {𝐴}
32preq2i 4249 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4 dfopg 4375 . . 3 ((𝐴𝑉𝐴𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
54anidms 676 . 2 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
6 dfsn2 4168 . . 3 {{𝐴}} = {{𝐴}, {𝐴}}
76a1i 11 . 2 (𝐴𝑉 → {{𝐴}} = {{𝐴}, {𝐴}})
83, 5, 73eqtr4a 2681 1 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {csn 4155  {cpr 4157  cop 4161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162
This theorem is referenced by: (None)
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