ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmsnsnsng Structured version   GIF version

Theorem dmsnsnsng 4741
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 2554 . . . . . . 7 x V
21opid 3558 . . . . . 6 x, x⟩ = {{x}}
3 sneq 3378 . . . . . . 7 (x = A → {x} = {A})
43sneqd 3380 . . . . . 6 (x = A → {{x}} = {{A}})
52, 4syl5eq 2081 . . . . 5 (x = A → ⟨x, x⟩ = {{A}})
65sneqd 3380 . . . 4 (x = A → {⟨x, x⟩} = {{{A}}})
76dmeqd 4480 . . 3 (x = A → dom {⟨x, x⟩} = dom {{{A}}})
87, 3eqeq12d 2051 . 2 (x = A → (dom {⟨x, x⟩} = {x} ↔ dom {{{A}}} = {A}))
91dmsnop 4737 . 2 dom {⟨x, x⟩} = {x}
108, 9vtoclg 2607 1 (A V → dom {{{A}}} = {A})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  cop 3370  dom cdm 4288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator