Theorem vtxval3sn 29014

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 4843	. . . 4 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
3	2	eqcomi 2739	. . 3 ⊢ {{𝐴}} = ⟨𝐴, 𝐴⟩
4	3	sneqi 4585	. 2 ⊢ {{{𝐴}}} = {⟨𝐴, 𝐴⟩}
5	1, 4	vtxvalsnop 29012	1 ⊢ (Vtx‘{{{𝐴}}}) = {𝐴}

Syntax hints:   = wceq 1541   ∈ wcel 2110  Vcvv 3434  {csn 4574  ⟨cop 4580  ‘cfv 6477  Vtxcvtx 28967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6433  df-fun 6479  df-fv 6485  df-1st 7916  df-vtx 28969

This theorem is referenced by: (None)