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Theorem dmsnsnsng 4826
Description: The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
Assertion
Ref Expression
dmsnsnsng (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})

Proof of Theorem dmsnsnsng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 2577 . . . . . . 7 𝑥 ∈ V
21opid 3595 . . . . . 6 𝑥, 𝑥⟩ = {{𝑥}}
3 sneq 3414 . . . . . . 7 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43sneqd 3416 . . . . . 6 (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
52, 4syl5eq 2100 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
65sneqd 3416 . . . 4 (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
76dmeqd 4565 . . 3 (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
87, 3eqeq12d 2070 . 2 (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
91dmsnop 4822 . 2 dom {⟨𝑥, 𝑥⟩} = {𝑥}
108, 9vtoclg 2630 1 (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3403  cop 3406  dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-dm 4383
This theorem is referenced by: (None)
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