Theorem uvtxel 29347

Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) (Revised by AV, 14-Feb-2022.)

Hypothesis

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxel	⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))

Distinct variable groups:   𝑛,𝐺   𝑛,𝑁   𝑛,𝑉

Proof of Theorem uvtxel

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4646	. . . 4 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁})
2	1	difeq2d 4131	. . 3 ⊢ (𝑣 = 𝑁 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑁}))
3		oveq2 7438	. . . 4 ⊢ (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
4	3	eleq2d 2815	. . 3 ⊢ (𝑣 = 𝑁 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
5	2, 4	raleqbidv 3339	. 2 ⊢ (𝑣 = 𝑁 → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
6		uvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
7	6	uvtxval 29346	. 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
8	5, 7	elrab2 3696	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))

Syntax hints:   ↔ wb 205   ∧ wa 394   = 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:  uvtxisvtx  29348  vtxnbuvtx  29350  uvtx2vtx1edg  29357  uvtx2vtx1edgb  29358  uvtxnbgrb  29360  iscplgrnb  29375  cplgr1v  29389  cusgrexi  29402  structtocusgr  29405