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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 (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁))

Proof of Theorem nbgrnself2OLD
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (𝑣 = 𝑁𝑣 = 𝑁)
2 oveq2 6804 . . . . 5 (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
31, 2neleq12d 3050 . . . 4 (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
43rspccv 3457 . . 3 (∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣) → (𝑁 ∈ (Vtx‘𝐺) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
5 eqid 2771 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
65nbgrnself 26478 . . . 4 𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣)
76a1i 11 . . 3 (𝐺𝑊 → ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣))
84, 7syl11 33 . 2 (𝑁 ∈ (Vtx‘𝐺) → (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
95nbgrisvtxOLD 26460 . . . . 5 ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑁 ∈ (Vtx‘𝐺))
109ex 397 . . . 4 (𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺)))
1110con3rr3 152 . . 3 𝑁 ∈ (Vtx‘𝐺) → (𝐺𝑊 → ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁)))
12 df-nel 3047 . . 3 (𝑁 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁))
1311, 12syl6ibr 242 . 2 𝑁 ∈ (Vtx‘𝐺) → (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
148, 13pm2.61i 176 1 (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁))
 Colors of variables: wff setvar class 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
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