Theorem opidg 4823

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4824. (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 4802	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
2	1	anidms 571	. 2 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3		dfsn2 4568	. . . . 5 ⊢ {𝐴} = {𝐴, 𝐴}
4	3	eqcomi 2748	. . . 4 ⊢ {𝐴, 𝐴} = {𝐴}
5	4	preq2i 4669	. . 3 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6		dfsn2 4568	. . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	5, 6	eqtr4i 2765	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
8	2, 7	eqtrdi 2790	1 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {csn 4555  {cpr 4557  ⟨cop 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562

This theorem is referenced by:  opid  4824  brin3  38806