Theorem nbgrnself2 40585

Description: A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)

Assertion

Ref	Expression
nbgrnself2	⊢ (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))

Proof of Theorem nbgrnself2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	nbgrnself 40583	. . . . 5 ⊢ ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣)
3	2	a1i 11	. . . 4 ⊢ (𝐺 ∈ 𝑊 → ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣))
4		id 22	. . . . . 6 ⊢ (𝑣 = 𝑁 → 𝑣 = 𝑁)
5		oveq2 6557	. . . . . 6 ⊢ (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
6	4, 5	neleq12d 2887	. . . . 5 ⊢ (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
7	6	rspccv 3279	. . . 4 ⊢ (∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣) → (𝑁 ∈ (Vtx‘𝐺) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
8	3, 7	syl 17	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (Vtx‘𝐺) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
9	8	com12 32	. 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
10	1	nbgrisvtx 40581	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑁 ∈ (Vtx‘𝐺))
11	10	ex 449	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺)))
12	11	con3rr3 150	. . 3 ⊢ (¬ 𝑁 ∈ (Vtx‘𝐺) → (𝐺 ∈ 𝑊 → ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁)))
13		df-nel 2783	. . 3 ⊢ (𝑁 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁))
14	12, 13	syl6ibr 241	. 2 ⊢ (¬ 𝑁 ∈ (Vtx‘𝐺) → (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
15	9, 14	pm2.61i 175	1 ⊢ (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  ‘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:  nbgrssovtx  40586  usgrnbnself2  40588