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	⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Proof of Theorem opidg

Step	Hyp	Ref	Expression
1		dfopg 4708	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
2	1	anidms 567	. 2 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3		dfsn2 4485	. . . . 5 ⊢ {𝐴} = {𝐴, 𝐴}
4	3	eqcomi 2804	. . . 4 ⊢ {𝐴, 𝐴} = {𝐴}
5	4	preq2i 4580	. . 3 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6		dfsn2 4485	. . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	5, 6	eqtr4i 2822	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
8	2, 7	syl6eq 2847	1 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

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