Theorem nbgrnself2 29382

Description: A class 𝑋 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.) (Revised by AV, 12-Feb-2022.)

Assertion

Ref	Expression
nbgrnself2	⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)

Proof of Theorem nbgrnself2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑣 = 𝑋 → 𝑣 = 𝑋)
2		oveq2 7364	. . . 4 ⊢ (𝑣 = 𝑋 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑋))
3	1, 2	neleq12d 3039	. . 3 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
4		eqid 2734	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	nbgrnself 29381	. . 3 ⊢ ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣)
6	3, 5	vtoclri 3542	. 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
7	4	nbgrisvtx 29363	. . . 4 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))
8	7	con3i 154	. . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
9		df-nel 3035	. . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
10	8, 9	sylibr 234	. 2 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
11	6, 10	pm2.61i 182	1 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113   ∉ wnel 3034  ‘cfv 6490  (class class class)co 7356  Vtxcvtx 29018   NeighbVtx cnbgr 29354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-nbgr 29355

This theorem is referenced by:  nbgrssovtx  29383  nb3grprlem2  29403  isubgr3stgrlem1  48154  isubgr3stgrlem3  48156