Theorem vtxval3sn 29678

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 4858	. . . 4 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
3	2	eqcomi 2772	. . 3 ⊢ {{𝐴}} = ⟨𝐴, 𝐴⟩
4	3	sneqi 4595	. 2 ⊢ {{{𝐴}}} = {⟨𝐴, 𝐴⟩}
5	1, 4	vtxvalsnop 29676	1 ⊢ (Vtx‘{{{𝐴}}}) = {𝐴}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588  ⟨cop 4840  ‘cfv 6566  Vtxcvtx 29631

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-1st 8035  df-vtx 29633

This theorem is used by: (None)