Theorem uvtxel 27178

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 4535	. . . 4 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁})
2	1	difeq2d 4050	. . 3 ⊢ (𝑣 = 𝑁 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑁}))
3		oveq2 7143	. . . 4 ⊢ (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
4	3	eleq2d 2875	. . 3 ⊢ (𝑣 = 𝑁 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
5	2, 4	raleqbidv 3354	. 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 27177	. 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
8	5, 7	elrab2 3631	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∖ cdif 3878  {csn 4525  ‘cfv 6324  (class class class)co 7135  Vtxcvtx 26789   NeighbVtx cnbgr 27122  UnivVtxcuvtx 27175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-uvtx 27176

This theorem is referenced by:  uvtxisvtx  27179  vtxnbuvtx  27181  uvtx2vtx1edg  27188  uvtx2vtx1edgb  27189  uvtxnbgrb  27191  iscplgrnb  27206  cplgr1v  27220  cusgrexi  27233  structtocusgr  27236