Theorem nbgrnself2OLD 26482

Description: Obsolete version of nbgrnself2 26479 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
nbgrnself2OLD	⊢ (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))

Proof of Theorem nbgrnself2OLD

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝑣 = 𝑁 → 𝑣 = 𝑁)
2		oveq2 6804	. . . . 5 ⊢ (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
3	1, 2	neleq12d 3050	. . . 4 ⊢ (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
4	3	rspccv 3457	. . 3 ⊢ (∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣) → (𝑁 ∈ (Vtx‘𝐺) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
5		eqid 2771	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6	5	nbgrnself 26478	. . . 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	nbgrisvtxOLD 26460	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑁 ∈ (Vtx‘𝐺))
10	9	ex 397	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺)))
11	10	con3rr3 152	. . 3 ⊢ (¬ 𝑁 ∈ (Vtx‘𝐺) → (𝐺 ∈ 𝑊 → ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁)))
12		df-nel 3047	. . 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 1631   ∈ wcel 2145   ∉ wnel 3046  ∀wral 3061  ‘cfv 6030  (class class class)co 6796  Vtxcvtx 26095   NeighbVtx cnbgr 26447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-nbgr 26448

This theorem is referenced by:  nbgrssovtxOLD  26483  usgrnbnself2OLD  26485