MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opidg Structured version   Visualization version   GIF version

Theorem opidg 4729
Description: The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Closed form of opid 4730. (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 4708 . . 3 ((𝐴𝑉𝐴𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
21anidms 567 . 2 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3 dfsn2 4485 . . . . 5 {𝐴} = {𝐴, 𝐴}
43eqcomi 2804 . . . 4 {𝐴, 𝐴} = {𝐴}
54preq2i 4580 . . 3 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6 dfsn2 4485 . . 3 {{𝐴}} = {{𝐴}, {𝐴}}
75, 6eqtr4i 2822 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
82, 7syl6eq 2847 1 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1522  wcel 2081  {csn 4472  {cpr 4474  cop 4478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479
This theorem is referenced by:  opid  4730  brin3  35208
  Copyright terms: Public domain W3C validator