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Theorem dmsnsnsng 5016
 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 2689 . . . . . . 7 𝑥 ∈ V
21opid 3723 . . . . . 6 𝑥, 𝑥⟩ = {{𝑥}}
3 sneq 3538 . . . . . . 7 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43sneqd 3540 . . . . . 6 (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
52, 4syl5eq 2184 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
65sneqd 3540 . . . 4 (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
76dmeqd 4741 . . 3 (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
87, 3eqeq12d 2154 . 2 (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
91dmsnop 5012 . 2 dom {⟨𝑥, 𝑥⟩} = {𝑥}
108, 9vtoclg 2746 1 (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2686  {csn 3527  ⟨cop 3530  dom cdm 4539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-dm 4549 This theorem is referenced by: (None)
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