Theorem opidg 40316

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4359. (Contributed by AV, 18-Sep-2020.) (Revised by AV, 18-Sep-2021.)

Assertion

Ref	Expression
opidg	⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Proof of Theorem opidg

Step	Hyp	Ref	Expression
1		dfsn2 4138	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	eqcomi 2619	. . 3 ⊢ {𝐴, 𝐴} = {𝐴}
3	2	preq2i 4216	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4		dfopg 4338	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
5	4	anidms 675	. 2 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
6		dfsn2 4138	. . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	6	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → {{𝐴}} = {{𝐴}, {𝐴}})
8	3, 5, 7	3eqtr4a 2670	1 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {csn 4125  {cpr 4127  ⟨cop 4131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132

This theorem is referenced by: (None)