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Theorem opidg 4642
 Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4643. (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 4621 . . 3 ((𝐴𝑉𝐴𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
21anidms 564 . 2 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3 dfsn2 4410 . . . . 5 {𝐴} = {𝐴, 𝐴}
43eqcomi 2834 . . . 4 {𝐴, 𝐴} = {𝐴}
54preq2i 4490 . . 3 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6 dfsn2 4410 . . 3 {{𝐴}} = {{𝐴}, {𝐴}}
75, 6eqtr4i 2852 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
82, 7syl6eq 2877 1 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  {csn 4397  {cpr 4399  ⟨cop 4403 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404 This theorem is referenced by:  opid  4643  brin3  34716
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