Theorem dmsnsnsn 5531

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 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
2	1	opid 4359	. . . . . . 7 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
3		sneq 4135	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	sneqd 4137	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
5	2, 4	syl5eq 2656	. . . . . 6 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
6	5	sneqd 4137	. . . . 5 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
7	6	dmeqd 5248	. . . 4 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
8	7, 3	eqeq12d 2625	. . 3 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
9	1	dmsnop 5527	. . 3 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
10	8, 9	vtoclg 3239	. 2 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11		0ex 4718	. . . . 5 ⊢ ∅ ∈ V
12	11	snid 4155	. . . 4 ⊢ ∅ ∈ {∅}
13		dmsn0el 5522	. . . 4 ⊢ (∅ ∈ {∅} → dom {{∅}} = ∅)
14	12, 13	ax-mp 5	. . 3 ⊢ dom {{∅}} = ∅
15		snprc 4197	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 205	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	16	sneqd 4137	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {{𝐴}} = {∅})
18	17	sneqd 4137	. . . 4 ⊢ (¬ 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
19	18	dmeqd 5248	. . 3 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
20	14, 19, 16	3eqtr4a 2670	. 2 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
21	10, 20	pm2.61i 175	1 ⊢ dom {{{𝐴}}} = {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131  dom cdm 5038

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

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-ne 2782  df-rab 2905  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  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048

This theorem is referenced by: (None)