Theorem uvtxnbgrvtx 29854

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 29852	. . 3 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
3		eleq1w 2843	. . . . . . 7 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
4	3	rspcva 3574	. . . . . 6 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
5		nbgrsym 29824	. . . . . . 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 3153	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 2145  ∀wral 3076   ∖ cdif 3896  {csn 4584  ‘cfv 6533  (class class class)co 7414  Vtxcvtx 29454   NeighbVtx cnbgr 29793  UnivVtxcuvtx 29846

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-nbgr 29794  df-uvtx 29847

This theorem is used by: (None)