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Theorem nbgrssovtx 40581
Description: The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 40577. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
Hypothesis
Ref Expression
nbgrssovtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrssovtx (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Proof of Theorem nbgrssovtx
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nbgrssovtx.v . . . . 5 𝑉 = (Vtx‘𝐺)
21nbgrisvtx 40576 . . . 4 ((𝐺𝑊𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣𝑉)
3 nbgrnself2 40580 . . . . . . . . . 10 (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁))
43adantr 479 . . . . . . . . 9 ((𝐺𝑊𝑣 = 𝑁) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
5 df-nel 2782 . . . . . . . . . 10 (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
6 neleq1 2887 . . . . . . . . . . 11 (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
76adantl 480 . . . . . . . . . 10 ((𝐺𝑊𝑣 = 𝑁) → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
85, 7syl5bbr 272 . . . . . . . . 9 ((𝐺𝑊𝑣 = 𝑁) → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
94, 8mpbird 245 . . . . . . . 8 ((𝐺𝑊𝑣 = 𝑁) → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
109ex 448 . . . . . . 7 (𝐺𝑊 → (𝑣 = 𝑁 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
1110con2d 127 . . . . . 6 (𝐺𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → ¬ 𝑣 = 𝑁))
1211imp 443 . . . . 5 ((𝐺𝑊𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → ¬ 𝑣 = 𝑁)
1312neqned 2788 . . . 4 ((𝐺𝑊𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣𝑁)
14 eldifsn 4259 . . . 4 (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣𝑉𝑣𝑁))
152, 13, 14sylanbrc 694 . . 3 ((𝐺𝑊𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝑉 ∖ {𝑁}))
1615ex 448 . 2 (𝐺𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
1716ssrdv 3573 1 (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wnel 2780  cdif 3536  wss 3539  {csn 4124  cfv 5790  (class class class)co 6527  Vtxcvtx 40224   NeighbVtx cnbgr 40545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-nbgr 40549
This theorem is referenced by:  nbgrssvwo2  40582  usgrnbssovtx  40584  nbfusgrlevtxm1  40600  uvtxnbgr  40622  nbusgrvtxm1uvtx  40627  nbupgruvtxres  40629
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