Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nbgrssvwo2 Structured version   Visualization version   GIF version

 Description: The neighbors of a vertex are a subset of all vertices except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
Hypothesis
Ref Expression
Assertion
Ref Expression
nbgrssvwo2 ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))

StepHypRef Expression
1 nbgrssovtx.v . . . . 5 𝑉 = (Vtx‘𝐺)
21nbgrssovtx 40586 . . . 4 (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
3 df-nel 2783 . . . . . 6 (𝑀 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑁))
4 disjsn 4192 . . . . . 6 (((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑁))
53, 4sylbb2 227 . . . . 5 (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → ((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅)
6 reldisj 3972 . . . . 5 ((𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
75, 6syl5ib 233 . . . 4 ((𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
82, 7syl 17 . . 3 (𝐺𝑊 → (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
98imp 444 . 2 ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))
10 prcom 4211 . . . 4 {𝑀, 𝑁} = {𝑁, 𝑀}
1110difeq2i 3687 . . 3 (𝑉 ∖ {𝑀, 𝑁}) = (𝑉 ∖ {𝑁, 𝑀})
12 difpr 4275 . . 3 (𝑉 ∖ {𝑁, 𝑀}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀})
1311, 12eqtri 2632 . 2 (𝑉 ∖ {𝑀, 𝑁}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀})
149, 13syl6sseqr 3615 1 ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   NeighbVtx cnbgr 40550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-nbgr 40554 This theorem is referenced by:  usgrnbssvwo2  40590  nbfusgrlevtxm2  40606
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