Theorem clnbgrn0 48320

Description: The closed neighborhood of a vertex is never empty. (Contributed by AV, 16-May-2025.)

Hypothesis

Ref	Expression
clnbgrn0.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
clnbgrn0	⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)

Proof of Theorem clnbgrn0

Step	Hyp	Ref	Expression
1		clnbgrn0.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	clnbgrvtxel 48317	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑁))
3		ne0i 4282	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)
4	2, 3	syl 17	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4274  ‘cfv 6492  (class class class)co 7360  Vtxcvtx 29079   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  ax-un 7682

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-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-clnbgr 48307

This theorem is referenced by: (None)