Theorem nbgrnself2 29447

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 3043	. . 3 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
4		eqid 2739	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	nbgrnself 29446	. . 3 ⊢ ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣)
6	3, 5	vtoclri 3528	. 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
7	4	nbgrisvtx 29428	. . . 4 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))
8	7	con3i 154	. . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
9		df-nel 3039	. . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
10	8, 9	sylibr 235	. 2 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
11	6, 10	pm2.61i 183	1 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119   ∉ wnel 3038  ‘cfv 6485  (class class class)co 7356  Vtxcvtx 29083   NeighbVtx cnbgr 29419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-nbgr 29420

This theorem is referenced by:  nbgrssovtx  29448  nb3grprlem2  29468  isubgr3stgrlem1  48457  isubgr3stgrlem3  48459