Theorem nbgrssovtx 40586

Description: The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 40582. (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 40581	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ 𝑉)
3		nbgrnself2 40585	. . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
4	3	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
5		df-nel 2783	. . . . . . . . . 10 ⊢ (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
6		neleq1 2888	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
7	6	adantl 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
8	5, 7	syl5bbr 273	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
9	4, 8	mpbird 246	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
10	9	ex 449	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (𝑣 = 𝑁 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
11	10	con2d 128	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → ¬ 𝑣 = 𝑁))
12	11	imp 444	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → ¬ 𝑣 = 𝑁)
13	12	neqned 2789	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ≠ 𝑁)
14		eldifsn 4260	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁))
15	2, 13, 14	sylanbrc 695	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝑉 ∖ {𝑁}))
16	15	ex 449	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
17	16	ssrdv 3574	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   NeighbVtx cnbgr 40550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-nbgr 40554

This theorem is referenced by:  nbgrssvwo2  40587  usgrnbssovtx  40589  nbfusgrlevtxm1  40605  uvtxnbgr  40627  nbusgrvtxm1uvtx  40632  nbupgruvtxres  40634