Theorem dmsnsnsn 6076

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 3497	. . . . . . . 8 ⊢ 𝑥 ∈ V
2	1	opid 4822	. . . . . . 7 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
3		sneq 4576	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	sneqd 4578	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
5	2, 4	syl5eq 2868	. . . . . 6 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
6	5	sneqd 4578	. . . . 5 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
7	6	dmeqd 5773	. . . 4 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
8	7, 3	eqeq12d 2837	. . 3 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
9	1	dmsnop 6072	. . 3 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
10	8, 9	vtoclg 3567	. 2 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11		0ex 5210	. . . . 5 ⊢ ∅ ∈ V
12	11	snid 4600	. . . 4 ⊢ ∅ ∈ {∅}
13		dmsn0el 6067	. . . 4 ⊢ (∅ ∈ {∅} → dom {{∅}} = ∅)
14	12, 13	ax-mp 5	. . 3 ⊢ dom {{∅}} = ∅
15		snprc 4652	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 218	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	16	sneqd 4578	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {{𝐴}} = {∅})
18	17	sneqd 4578	. . . 4 ⊢ (¬ 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
19	18	dmeqd 5773	. . 3 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
20	14, 19, 16	3eqtr4a 2882	. 2 ⊢ (¬ 𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
21	10, 20	pm2.61i 184	1 ⊢ dom {{{𝐴}}} = {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  {csn 4566  ⟨cop 4572  dom cdm 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-dm 5564

This theorem is referenced by: (None)