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Theorem dmsnsnsn 5649
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.
StepHypRef Expression
1 vex 3234 . . . . . . . 8 𝑥 ∈ V
21opid 4453 . . . . . . 7 𝑥, 𝑥⟩ = {{𝑥}}
3 sneq 4220 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43sneqd 4222 . . . . . . 7 (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
52, 4syl5eq 2697 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
65sneqd 4222 . . . . 5 (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
76dmeqd 5358 . . . 4 (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
87, 3eqeq12d 2666 . . 3 (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
91dmsnop 5645 . . 3 dom {⟨𝑥, 𝑥⟩} = {𝑥}
108, 9vtoclg 3297 . 2 (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11 0ex 4823 . . . . 5 ∅ ∈ V
1211snid 4241 . . . 4 ∅ ∈ {∅}
13 dmsn0el 5639 . . . 4 (∅ ∈ {∅} → dom {{∅}} = ∅)
1412, 13ax-mp 5 . . 3 dom {{∅}} = ∅
15 snprc 4285 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 206 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
1716sneqd 4222 . . . . 5 𝐴 ∈ V → {{𝐴}} = {∅})
1817sneqd 4222 . . . 4 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
1918dmeqd 5358 . . 3 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
2014, 19, 163eqtr4a 2711 . 2 𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
2110, 20pm2.61i 176 1 dom {{{𝐴}}} = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  {csn 4210  cop 4216  dom cdm 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-dm 5153
This theorem is referenced by: (None)
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