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

Theorem dmsnsnsn 6050
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 3474 . . . . . . . 8 𝑥 ∈ V
21opid 4796 . . . . . . 7 𝑥, 𝑥⟩ = {{𝑥}}
3 sneq 4550 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43sneqd 4552 . . . . . . 7 (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
52, 4syl5eq 2868 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
65sneqd 4552 . . . . 5 (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
76dmeqd 5747 . . . 4 (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
87, 3eqeq12d 2837 . . 3 (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
91dmsnop 6046 . . 3 dom {⟨𝑥, 𝑥⟩} = {𝑥}
108, 9vtoclg 3544 . 2 (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
11 0ex 5184 . . . . 5 ∅ ∈ V
1211snid 4574 . . . 4 ∅ ∈ {∅}
13 dmsn0el 6041 . . . 4 (∅ ∈ {∅} → dom {{∅}} = ∅)
1412, 13ax-mp 5 . . 3 dom {{∅}} = ∅
15 snprc 4626 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 219 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
1716sneqd 4552 . . . . 5 𝐴 ∈ V → {{𝐴}} = {∅})
1817sneqd 4552 . . . 4 𝐴 ∈ V → {{{𝐴}}} = {{∅}})
1918dmeqd 5747 . . 3 𝐴 ∈ V → dom {{{𝐴}}} = dom {{∅}})
2014, 19, 163eqtr4a 2882 . 2 𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
2110, 20pm2.61i 185 1 dom {{{𝐴}}} = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2115  Vcvv 3471  c0 4266  {csn 4540  cop 4546  dom cdm 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-xp 5534  df-dm 5538
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator