Theorem opidg 41297

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4421. (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 4190	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	eqcomi 2631	. . 3 ⊢ {𝐴, 𝐴} = {𝐴}
3	2	preq2i 4272	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4		dfopg 4400	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
5	4	anidms 677	. 2 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
6		dfsn2 4190	. . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	6	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → {{𝐴}} = {{𝐴}, {𝐴}})
8	3, 5, 7	3eqtr4a 2682	1 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {csn 4177  {cpr 4179  ⟨cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by: (None)