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Theorem nbgrssovtx 26302
Description: The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself. Stronger version of nbgrssvtx 26281. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.)
Hypothesis
Ref Expression
nbgrssovtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrssovtx (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})

Proof of Theorem nbgrssovtx
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nbgrssovtx.v . . . 4 𝑉 = (Vtx‘𝐺)
21nbgrisvtx 26280 . . 3 (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣𝑉)
3 nbgrnself2 26301 . . . . 5 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
4 df-nel 2927 . . . . . 6 (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
5 neleq1 2931 . . . . . 6 (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
64, 5syl5bbr 274 . . . . 5 (𝑣 = 𝑋 → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
73, 6mpbiri 248 . . . 4 (𝑣 = 𝑋 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
87necon2ai 2852 . . 3 (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣𝑋)
9 eldifsn 4350 . . 3 (𝑣 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑣𝑉𝑣𝑋))
102, 8, 9sylanbrc 699 . 2 (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ (𝑉 ∖ {𝑋}))
1110ssriv 3640 1 (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  wne 2823  wnel 2926  cdif 3604  wss 3607  {csn 4210  cfv 5926  (class class class)co 6690  Vtxcvtx 25919   NeighbVtx cnbgr 26269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-nbgr 26270
This theorem is referenced by:  nbgrssvwo2  26303  nbfusgrlevtxm1  26323  uvtxnbgr  26351  nbusgrvtxm1uvtx  26356  nbupgruvtxres  26358
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