Theorem clnbgrcl 47808

Description: If a class 𝑋 has at least one element in its closed neighborhood, this class must be a vertex. (Contributed by AV, 7-May-2025.)

Hypothesis

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

Assertion

Ref	Expression
clnbgrcl	⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑋 ∈ 𝑉)

Proof of Theorem clnbgrcl

Dummy variables 𝑔 𝑒 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clnbgr 47806	. . 3 ⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
2	1	mpoxeldm 8236	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → (𝐺 ∈ V ∧ 𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
3		csbfv 6956	. . . . 5 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
4		clnbgrcl.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	eqtr4i 2768	. . . 4 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = 𝑉
6	5	eleq2i 2833	. . 3 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) ↔ 𝑋 ∈ 𝑉)
7	6	biimpi 216	. 2 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → 𝑋 ∈ 𝑉)
8	2, 7	simpl2im 503	1 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑋 ∈ 𝑉)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  {crab 3436  Vcvv 3480  ⦋csb 3899   ∪ cun 3949   ⊆ wss 3951  {csn 4626  {cpr 4628  ‘cfv 6561  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064   ClNeighbVtx cclnbgr 47805

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-clnbgr 47806

This theorem is referenced by:  elclnbgrelnbgr  47812  clnbgrel  47815