Theorem opidg 4775

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4776. (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 4752	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
2	1	anidms 571	. 2 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}})
3		dfsn2 4528	. . . . 5 ⊢ {𝐴} = {𝐴, 𝐴}
4	3	eqcomi 2768	. . . 4 ⊢ {𝐴, 𝐴} = {𝐴}
5	4	preq2i 4623	. . 3 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
6		dfsn2 4528	. . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	5, 6	eqtr4i 2785	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}}
8	2, 7	eqtrdi 2810	1 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  {csn 4515  {cpr 4517  ⟨cop 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3409  df-dif 3857  df-un 3859  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522

This theorem is referenced by:  opid  4776  brin3  36083