Theorem uvtxnbgrvtx 27183

Description: A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)

Hypothesis

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxnbgrvtx	⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣))

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

Proof of Theorem uvtxnbgrvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxel.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	vtxnbuvtx 27181	. . 3 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
3		eleq1w 2872	. . . . . . 7 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
4	3	rspcva 3569	. . . . . 6 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
5		nbgrsym 27153	. . . . . . 7 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∈ (𝐺 NeighbVtx 𝑣))
6	5	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))
7	4, 6	syl5ibcom 248	. . . . 5 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) → (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))
8	7	expcom 417	. . . 4 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣))))
9	8	com23 86	. . 3 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁) → (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣))))
10	2, 9	mpcom 38	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))
11	10	ralrimiv 3148	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣))

Syntax hints:   → wi 4   ↔ 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  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  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-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  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-iun 4883  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-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-nbgr 27123  df-uvtx 27176

This theorem is referenced by: (None)