Theorem clnbgrprc0 48308

Description: The closed neighborhood is empty if the graph 𝐺 or the vertex 𝑁 are proper classes. (Contributed by AV, 7-May-2025.)

Assertion

Ref	Expression
clnbgrprc0	⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 ClNeighbVtx 𝑁) = ∅)

Proof of Theorem clnbgrprc0

Dummy variables 𝑒 𝑔 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clnbgr 48307	. . 3 ⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
2	1	reldmmpo 7494	. 2 ⊢ Rel dom ClNeighbVtx
3	2	ovprc 7398	1 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 ClNeighbVtx 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3390  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  {csn 4568  {cpr 4570  ‘cfv 6492  (class class class)co 7360  Vtxcvtx 29079  Edgcedg 29130   ClNeighbVtx cclnbgr 48306

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-dm 5634  df-iota 6448  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-clnbgr 48307

This theorem is referenced by: (None)