Theorem nbgrnself2 26259

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		id 22	. . . . 5 ⊢ (𝑣 = 𝑁 → 𝑣 = 𝑁)
2		oveq2 6658	. . . . 5 ⊢ (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
3	1, 2	neleq12d 2901	. . . 4 ⊢ (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
4	3	rspccv 3306	. . 3 ⊢ (∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣) → (𝑁 ∈ (Vtx‘𝐺) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
5		eqid 2622	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6	5	nbgrnself 26257	. . . 4 ⊢ ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣)
7	6	a1i 11	. . 3 ⊢ (𝐺 ∈ 𝑊 → ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣))
8	4, 7	syl11 33	. 2 ⊢ (𝑁 ∈ (Vtx‘𝐺) → (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
9	5	nbgrisvtx 26255	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑁 ∈ (Vtx‘𝐺))
10	9	ex 450	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺)))
11	10	con3rr3 151	. . 3 ⊢ (¬ 𝑁 ∈ (Vtx‘𝐺) → (𝐺 ∈ 𝑊 → ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁)))
12		df-nel 2898	. . 3 ⊢ (𝑁 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁))
13	11, 12	syl6ibr 242	. 2 ⊢ (¬ 𝑁 ∈ (Vtx‘𝐺) → (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
14	8, 13	pm2.61i 176	1 ⊢ (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  ∀wral 2912  ‘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:  nbgrssovtx  26260  usgrnbnself2  26262