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Theorem vtxval3sn 25718
 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.)
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 4359 . . . 4 𝐴, 𝐴⟩ = {{𝐴}}
32eqcomi 2619 . . 3 {{𝐴}} = ⟨𝐴, 𝐴
43sneqi 4136 . 2 {{{𝐴}}} = {⟨𝐴, 𝐴⟩}
51, 4vtxvalsnop 25716 1 (Vtx‘{{{𝐴}}}) = {𝐴}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  ‘cfv 5804  Vtxcvtx 25673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-vtx 25675 This theorem is referenced by: (None)
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