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Theorem dmsnsnsn 6176
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 3451 . . . . . . . 8 𝑥 ∈ V
21opid 4854 . . . . . . 7 𝑥, 𝑥⟩ = {{𝑥}}
3 sneq 4600 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43sneqd 4602 . . . . . . 7 (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
52, 4eqtrid 2785 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
65sneqd 4602 . . . . 5 (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
76dmeqd 5865 . . . 4 (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
87, 3eqeq12d 2749 . . 3 (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
91dmsnop 6172 . . 3 dom {⟨𝑥, 𝑥⟩} = {𝑥}
108, 9vtoclg 3527 . 2 (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11 0ex 5268 . . . . 5 ∅ ∈ V
1211snid 4626 . . . 4 ∅ ∈ {∅}
13 dmsn0el 6167 . . . 4 (∅ ∈ {∅} → dom {{∅}} = ∅)
1412, 13ax-mp 5 . . 3 dom {{∅}} = ∅
15 snprc 4682 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 215 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
1716sneqd 4602 . . . . 5 𝐴 ∈ V → {{𝐴}} = {∅})
1817sneqd 4602 . . . 4 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
1918dmeqd 5865 . . 3 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
2014, 19, 163eqtr4a 2799 . 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 1542  wcel 2107  Vcvv 3447  c0 4286  {csn 4590  cop 4596  dom cdm 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-dm 5647
This theorem is referenced by: (None)
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