Theorem nbgrssovtx 26260

Description: The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 26256. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)

Hypothesis

Ref	Expression
nbgrssovtx.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbgrssovtx	⊢ (𝐺 ∈ 𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Proof of Theorem nbgrssovtx

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgrssovtx.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrisvtx 26255	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ 𝑉)
3		nbgrnself2 26259	. . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
4	3	adantr 481	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
5		df-nel 2898	. . . . . . . . . 10 ⊢ (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
6		neleq1 2902	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
7	6	adantl 482	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
8	5, 7	syl5bbr 274	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
9	4, 8	mpbird 247	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
10	9	ex 450	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (𝑣 = 𝑁 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
11	10	con2d 129	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → ¬ 𝑣 = 𝑁))
12	11	imp 445	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → ¬ 𝑣 = 𝑁)
13	12	neqned 2801	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ≠ 𝑁)
14		eldifsn 4317	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁))
15	2, 13, 14	sylanbrc 698	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝑉 ∖ {𝑁}))
16	15	ex 450	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
17	16	ssrdv 3609	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∉ wnel 2897   ∖ cdif 3571   ⊆ wss 3574  {csn 4177  ‘cfv 5888  (class class class)co 6650  Vtxcvtx 25874   NeighbVtx cnbgr 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-nbgr 26228

This theorem is referenced by:  nbgrssvwo2  26261  usgrnbssovtx  26263  nbfusgrlevtxm1  26279  uvtxnbgr  26301  nbusgrvtxm1uvtx  26306  nbupgruvtxres  26308