Theorem dmsnsnsn 6177

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 3450	. . . . . . . 8 ⊢ 𝑥 ∈ V
2	1	opid 4855	. . . . . . 7 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
3		sneq 4601	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	sneqd 4603	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
5	2, 4	eqtrid 2783	. . . . . 6 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
6	5	sneqd 4603	. . . . 5 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
7	6	dmeqd 5866	. . . 4 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
8	7, 3	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
9	1	dmsnop 6173	. . 3 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
10	8, 9	vtoclg 3526	. 2 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11		0ex 5269	. . . . 5 ⊢ ∅ ∈ V
12	11	snid 4627	. . . 4 ⊢ ∅ ∈ {∅}
13		dmsn0el 6168	. . . 4 ⊢ (∅ ∈ {∅} → dom {{∅}} = ∅)
14	12, 13	ax-mp 5	. . 3 ⊢ dom {{∅}} = ∅
15		snprc 4683	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 215	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	16	sneqd 4603	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {{𝐴}} = {∅})
18	17	sneqd 4603	. . . 4 ⊢ (¬ 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
19	18	dmeqd 5866	. . 3 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
20	14, 19, 16	3eqtr4a 2797	. 2 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
21	10, 20	pm2.61i 182	1 ⊢ dom {{{𝐴}}} = {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  Vcvv 3446  ∅c0 4287  {csn 4591  ⟨cop 4597  dom cdm 5638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-dm 5648

This theorem is referenced by: (None)