Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbgrprc0 Structured version   Visualization version   GIF version

Theorem nbgrprc0 27115
 Description: The set of neighbors is empty if the graph 𝐺 or the vertex 𝑁 are proper classes. (Contributed by AV, 26-Oct-2020.)
Assertion
Ref Expression
nbgrprc0 (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)

Proof of Theorem nbgrprc0
Dummy variables 𝑒 𝑔 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgr 27114 . . 3 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
21reldmmpo 7269 . 2 Rel dom NeighbVtx
32ovprc 7178 1 (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∃wrex 3133  {crab 3136  Vcvv 3479   ∖ cdif 3915   ⊆ wss 3918  ∅c0 4274  {csn 4548  {cpr 4550  ‘cfv 6338  (class class class)co 7140  Vtxcvtx 26780  Edgcedg 26831   NeighbVtx cnbgr 27113 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-xp 5544  df-rel 5545  df-dm 5548  df-iota 6297  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-nbgr 27114 This theorem is referenced by:  uhgrnbgr0nb  27135  nbgr0vtxlem  27136
 Copyright terms: Public domain W3C validator