Theorem nbgrprc0 27931

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 27930	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
2	1	reldmmpo 7462	. 2 ⊢ Rel dom NeighbVtx
3	2	ovprc 7367	1 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∃wrex 3070  {crab 3403  Vcvv 3441   ∖ cdif 3894   ⊆ wss 3897  ∅c0 4268  {csn 4572  {cpr 4574  ‘cfv 6473  (class class class)co 7329  Vtxcvtx 27596  Edgcedg 27647   NeighbVtx cnbgr 27929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-dm 5624  df-iota 6425  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-nbgr 27930

This theorem is referenced by:  uhgrnbgr0nb  27951  nbgr0vtxlem  27952