Theorem uvtxisvtx 29348

Description: A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Proof shortened by AV, 14-Feb-2022.)

Hypothesis

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxisvtx	⊢ (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ 𝑉)

Proof of Theorem uvtxisvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	uvtxel 29347	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
3	2	simplbi 496	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ 𝑉)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2100  ∀wral 3054   ∖ cdif 3956  {csn 4636  ‘cfv 6558  (class class class)co 7430  Vtxcvtx 28955   NeighbVtx cnbgr 29291  UnivVtxcuvtx 29344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pr 5437

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-br 5157  df-opab 5219  df-mpt 5240  df-id 5584  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-iota 6510  df-fun 6560  df-fv 6566  df-ov 7433  df-uvtx 29345

This theorem is referenced by:  uvtxssvtx  29349  uvtxnm1nbgr  29363  vdiscusgr  29491