Theorem clnbgrprc0 47808

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 47807	. . 3 ⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
2	1	reldmmpo 7487	. 2 ⊢ Rel dom ClNeighbVtx
3	2	ovprc 7391	1 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 ClNeighbVtx 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3396  Vcvv 3438   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  {csn 4579  {cpr 4581  ‘cfv 6486  (class class class)co 7353  Vtxcvtx 28959  Edgcedg 29010   ClNeighbVtx cclnbgr 47806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-dm 5633  df-iota 6442  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-clnbgr 47807

This theorem is referenced by: (None)