Theorem nbgrprc0 29535

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 29534	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
2	1	reldmmpo 7530	. 2 ⊢ Rel dom NeighbVtx
3	2	ovprc 7434	1 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃wrex 3086  {crab 3414  Vcvv 3454   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  {csn 4582  {cpr 4584  ‘cfv 6521  (class class class)co 7396  Vtxcvtx 29197  Edgcedg 29248   NeighbVtx cnbgr 29533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-dm 5657  df-iota 6477  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-nbgr 29534

This theorem is referenced by:  uhgrnbgr0nb  29555  nbgr0edglem  29557