Theorem dmsnsnsn 6196

Description: The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
dmsnsnsn	⊢ dom {{{𝐴}}} = {𝐴}

Proof of Theorem dmsnsnsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
2	1	opid 4860	. . . . . . 7 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
3		sneq 4602	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	sneqd 4604	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
5	2, 4	eqtrid 2777	. . . . . 6 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
6	5	sneqd 4604	. . . . 5 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
7	6	dmeqd 5872	. . . 4 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
8	7, 3	eqeq12d 2746	. . 3 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
9	1	dmsnop 6192	. . 3 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
10	8, 9	vtoclg 3523	. 2 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11		0ex 5265	. . . . 5 ⊢ ∅ ∈ V
12	11	snid 4629	. . . 4 ⊢ ∅ ∈ {∅}
13		dmsn0el 6187	. . . 4 ⊢ (∅ ∈ {∅} → dom {{∅}} = ∅)
14	12, 13	ax-mp 5	. . 3 ⊢ dom {{∅}} = ∅
15		snprc 4684	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 216	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	16	sneqd 4604	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {{𝐴}} = {∅})
18	17	sneqd 4604	. . . 4 ⊢ (¬ 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
19	18	dmeqd 5872	. . 3 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
20	14, 19, 16	3eqtr4a 2791	. 2 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
21	10, 20	pm2.61i 182	1 ⊢ dom {{{𝐴}}} = {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  {csn 4592  ⟨cop 4598  dom cdm 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-dm 5651

This theorem is referenced by: (None)