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Theorem dmsnsnsn 6242
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 3482 . . . . . . . 8 𝑥 ∈ V
21opid 4898 . . . . . . 7 𝑥, 𝑥⟩ = {{𝑥}}
3 sneq 4641 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43sneqd 4643 . . . . . . 7 (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
52, 4eqtrid 2787 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
65sneqd 4643 . . . . 5 (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
76dmeqd 5919 . . . 4 (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
87, 3eqeq12d 2751 . . 3 (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
91dmsnop 6238 . . 3 dom {⟨𝑥, 𝑥⟩} = {𝑥}
108, 9vtoclg 3554 . 2 (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11 0ex 5313 . . . . 5 ∅ ∈ V
1211snid 4667 . . . 4 ∅ ∈ {∅}
13 dmsn0el 6233 . . . 4 (∅ ∈ {∅} → dom {{∅}} = ∅)
1412, 13ax-mp 5 . . 3 dom {{∅}} = ∅
15 snprc 4722 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 216 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
1716sneqd 4643 . . . . 5 𝐴 ∈ V → {{𝐴}} = {∅})
1817sneqd 4643 . . . 4 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
1918dmeqd 5919 . . 3 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
2014, 19, 163eqtr4a 2801 . 2 𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
2110, 20pm2.61i 182 1 dom {{{𝐴}}} = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631  cop 4637  dom cdm 5689
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699
This theorem is referenced by: (None)
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