MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmsnsnsn Structured version   Visualization version   GIF version

Theorem dmsnsnsn 6240
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 3484 . . . . . . . 8 𝑥 ∈ V
21opid 4893 . . . . . . 7 𝑥, 𝑥⟩ = {{𝑥}}
3 sneq 4636 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43sneqd 4638 . . . . . . 7 (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
52, 4eqtrid 2789 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
65sneqd 4638 . . . . 5 (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
76dmeqd 5916 . . . 4 (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
87, 3eqeq12d 2753 . . 3 (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
91dmsnop 6236 . . 3 dom {⟨𝑥, 𝑥⟩} = {𝑥}
108, 9vtoclg 3554 . 2 (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11 0ex 5307 . . . . 5 ∅ ∈ V
1211snid 4662 . . . 4 ∅ ∈ {∅}
13 dmsn0el 6231 . . . 4 (∅ ∈ {∅} → dom {{∅}} = ∅)
1412, 13ax-mp 5 . . 3 dom {{∅}} = ∅
15 snprc 4717 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 216 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
1716sneqd 4638 . . . . 5 𝐴 ∈ V → {{𝐴}} = {∅})
1817sneqd 4638 . . . 4 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
1918dmeqd 5916 . . 3 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
2014, 19, 163eqtr4a 2803 . 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 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626  cop 4632  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator