Theorem opidg 4835

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4836. (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 4814	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
2	1	anidms 566	. 2 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3		dfsn2 4580	. . . . 5 ⊢ {𝐴} = {𝐴, 𝐴}
4	3	eqcomi 2745	. . . 4 ⊢ {𝐴, 𝐴} = {𝐴}
5	4	preq2i 4681	. . 3 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6		dfsn2 4580	. . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	5, 6	eqtr4i 2762	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
8	2, 7	eqtrdi 2787	1 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4567  {cpr 4569  ⟨cop 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574

This theorem is referenced by:  opid  4836  brin3  38760