Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbgrssvwo2 Structured version   Visualization version   GIF version

 Description: The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself and a class 𝑀 which is not a neighbor of 𝑋. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.)
Hypothesis
Ref Expression
Assertion
Ref Expression
nbgrssvwo2 (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋}))

StepHypRef Expression
1 nbgrssovtx.v . . . 4 𝑉 = (Vtx‘𝐺)
21nbgrssovtx 27158 . . 3 (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})
3 df-nel 3092 . . . . 5 (𝑀 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
4 disjsn 4607 . . . . 5 (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
53, 4sylbb2 241 . . . 4 (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → ((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅)
6 reldisj 4359 . . . 4 ((𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) → (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀})))
75, 6syl5ib 247 . . 3 ((𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) → (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀})))
82, 7ax-mp 5 . 2 (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀}))
9 prcom 4628 . . . 4 {𝑀, 𝑋} = {𝑋, 𝑀}
109difeq2i 4047 . . 3 (𝑉 ∖ {𝑀, 𝑋}) = (𝑉 ∖ {𝑋, 𝑀})
11 difpr 4696 . . 3 (𝑉 ∖ {𝑋, 𝑀}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
1210, 11eqtri 2821 . 2 (𝑉 ∖ {𝑀, 𝑋}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
138, 12sseqtrrdi 3966 1 (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111   ∉ wnel 3091   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525  {cpr 4527  ‘cfv 6324  (class class class)co 7135  Vtxcvtx 26796   NeighbVtx cnbgr 27129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-nbgr 27130 This theorem is referenced by:  nbfusgrlevtxm2  27175
 Copyright terms: Public domain W3C validator