Theorem uvtxnbgrvtx 30031

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 30029	. . 3 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
3		eleq1w 2847	. . . . . . 7 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
4	3	rspcva 3578	. . . . . 6 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
5		nbgrsym 30001	. . . . . . 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 419	. . . 4 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣))))
9	8	com23 87	. . 3 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁) → (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣))))
10	2, 9	mpcom 39	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))
11	10	ralrimiv 3157	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∖ cdif 4048  {csn 4588  ‘cfv 6566  (class class class)co 7451  Vtxcvtx 29631   NeighbVtx cnbgr 29970  UnivVtxcuvtx 30023

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-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  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-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-nbgr 29971  df-uvtx 30024

This theorem is used by: (None)