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

Theorem nbgrssovtx 27155
Description: The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself. Stronger version of nbgrssvtx 27136. (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 27135 . . 3 (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣𝑉)
3 nbgrnself2 27154 . . . . 5 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
4 df-nel 3095 . . . . . 6 (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
5 neleq1 3099 . . . . . 6 (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
64, 5bitr3id 288 . . . . 5 (𝑣 = 𝑋 → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
73, 6mpbiri 261 . . . 4 (𝑣 = 𝑋 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
87necon2ai 3019 . . 3 (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣𝑋)
9 eldifsn 4683 . . 3 (𝑣 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑣𝑉𝑣𝑋))
102, 8, 9sylanbrc 586 . 2 (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ (𝑉 ∖ {𝑋}))
1110ssriv 3922 1 (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2112  wne 2990  wnel 3094  cdif 3881  wss 3884  {csn 4528  cfv 6328  (class class class)co 7139  Vtxcvtx 26793   NeighbVtx cnbgr 27126
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-nbgr 27127
This theorem is referenced by:  nbgrssvwo2  27156  nbfusgrlevtxm1  27171  uvtxnbgr  27194  nbusgrvtxm1uvtx  27199  nbupgruvtxres  27201
  Copyright terms: Public domain W3C validator