Theorem clnbgrprc0 48403

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 48402	. . 3 ⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
2	1	reldmmpo 7525	. 2 ⊢ Rel dom ClNeighbVtx
3	2	ovprc 7429	1 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 ClNeighbVtx 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  {crab 3413  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  ∅c0 4283  {csn 4579  {cpr 4581  ‘cfv 6516  (class class class)co 7391  Vtxcvtx 29154  Edgcedg 29205   ClNeighbVtx cclnbgr 48401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-dm 5653  df-iota 6472  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-clnbgr 48402

This theorem is referenced by: (None)