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Theorem nbgrprc0 26150
 Description: The set of neighbors is empty if the graph or the vertex 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 26149 . . 3 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
21reldmmpt2 6736 . 2 Rel dom NeighbVtx
32ovprc 6648 1 (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2909  {crab 2912  Vcvv 3190   ∖ cdif 3557   ⊆ wss 3560  ∅c0 3897  {csn 4155  {cpr 4157  ‘cfv 5857  (class class class)co 6615  Vtxcvtx 25808  Edgcedg 25873   NeighbVtx cnbgr 26145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-dm 5094  df-iota 5820  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-nbgr 26149 This theorem is referenced by:  uhgrnbgr0nb  26171  nbgr0vtxlem  26172
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