Theorem nbgrcl 27605

Description: If a class 𝑋 has at least one neighbor, this class must be a vertex. (Contributed by AV, 6-Jun-2021.) (Revised by AV, 12-Feb-2022.)

Hypothesis

Ref	Expression
nbgrcl.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbgrcl	⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)

Proof of Theorem nbgrcl

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

Step	Hyp	Ref	Expression
1		df-nbgr 27603	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
2	1	mpoxeldm 7998	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ V ∧ 𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
3		csbfv 6801	. . . . 5 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
4		nbgrcl.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	eqtr4i 2769	. . . 4 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = 𝑉
6	5	eleq2i 2830	. . 3 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) ↔ 𝑋 ∈ 𝑉)
7	6	biimpi 215	. 2 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → 𝑋 ∈ 𝑉)
8	2, 7	simpl2im 503	1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  {crab 3067  Vcvv 3422  ⦋csb 3828   ∖ cdif 3880   ⊆ wss 3883  {csn 4558  {cpr 4560  ‘cfv 6418  (class class class)co 7255  Vtxcvtx 27269  Edgcedg 27320   NeighbVtx cnbgr 27602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-nbgr 27603

This theorem is referenced by:  nbgrel  27610  frgrnbnb  28558