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Theorem vtxval3sn 29060
Description: Degenerated case 3 for vertices: The set of vertices of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.)
Hypothesis
Ref Expression
vtxval3sn.a 𝐴 ∈ V
Assertion
Ref Expression
vtxval3sn (Vtx‘{{{𝐴}}}) = {𝐴}

Proof of Theorem vtxval3sn
StepHypRef Expression
1 vtxval3sn.a . 2 𝐴 ∈ V
21opid 4893 . . . 4 𝐴, 𝐴⟩ = {{𝐴}}
32eqcomi 2746 . . 3 {{𝐴}} = ⟨𝐴, 𝐴
43sneqi 4637 . 2 {{{𝐴}}} = {⟨𝐴, 𝐴⟩}
51, 4vtxvalsnop 29058 1 (Vtx‘{{{𝐴}}}) = {𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632  cfv 6561  Vtxcvtx 29013
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-vtx 29015
This theorem is referenced by: (None)
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