Theorem clnbgrn0 48597

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 48594	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑁))
3		ne0i 4294	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)
4	2, 3	syl 18	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∅c0 4286  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29346   ClNeighbVtx cclnbgr 48583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-clnbgr 48584

This theorem is referenced by: (None)