Theorem nbgrprc0 27604

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.

Step	Hyp	Ref	Expression
1		df-nbgr 27603	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
2	1	reldmmpo 7386	. 2 ⊢ Rel dom NeighbVtx
3	2	ovprc 7293	1 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  {crab 3067  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  {csn 4558  {cpr 4560  ‘cfv 6418  (class class class)co 7255  Vtxcvtx 27269  Edgcedg 27320   NeighbVtx cnbgr 27602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-dm 5590  df-iota 6376  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-nbgr 27603

This theorem is referenced by:  uhgrnbgr0nb  27624  nbgr0vtxlem  27625