Theorem vtxval3sn 27316

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

Step	Hyp	Ref	Expression
1		vtxval3sn.a	. 2 ⊢ 𝐴 ∈ V
2	1	opid 4821	. . . 4 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
3	2	eqcomi 2747	. . 3 ⊢ {{𝐴}} = ⟨𝐴, 𝐴⟩
4	3	sneqi 4569	. 2 ⊢ {{{𝐴}}} = {⟨𝐴, 𝐴⟩}
5	1, 4	vtxvalsnop 27314	1 ⊢ (Vtx‘{{{𝐴}}}) = {𝐴}

Syntax hints:   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558  ⟨cop 4564  ‘cfv 6418  Vtxcvtx 27269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804  df-vtx 27271

This theorem is referenced by: (None)