Theorem clnbgrn0 47824

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 47821	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑁))
3		ne0i 4340	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)
4	2, 3	syl 17	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∅c0 4332  ‘cfv 6560  (class class class)co 7432  Vtxcvtx 29014   ClNeighbVtx cclnbgr 47810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-clnbgr 47811

This theorem is referenced by: (None)