Theorem opidg 4861

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4862. (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 4840	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
2	1	anidms 576	. 2 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3		dfsn2 4607	. . . . 5 ⊢ {𝐴} = {𝐴, 𝐴}
4	3	eqcomi 2778	. . . 4 ⊢ {𝐴, 𝐴} = {𝐴}
5	4	preq2i 4708	. . 3 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6		dfsn2 4607	. . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	5, 6	eqtr4i 2795	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
8	2, 7	eqtrdi 2820	1 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {csn 4594  {cpr 4596  ⟨cop 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601

This theorem is referenced by:  opid  4862  brin3  38977