Theorem clnbgrcl 47746

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 47744	. . 3 ⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
2	1	mpoxeldm 8235	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → (𝐺 ∈ V ∧ 𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
3		csbfv 6957	. . . . 5 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
4		clnbgrcl.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	eqtr4i 2766	. . . 4 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = 𝑉
6	5	eleq2i 2831	. . 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 1537   ∈ wcel 2106  ∃wrex 3068  {crab 3433  Vcvv 3478  ⦋csb 3908   ∪ cun 3961   ⊆ wss 3963  {csn 4631  {cpr 4633  ‘cfv 6563  (class class class)co 7431  Vtxcvtx 29028  Edgcedg 29079   ClNeighbVtx cclnbgr 47743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-clnbgr 47744

This theorem is referenced by:  elclnbgrelnbgr  47750  clnbgrel  47753