Theorem nbgrcl 29625

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 29623	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
2	1	mpoxeldm 8206	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ V ∧ 𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
3		csbfv 6929	. . . . 5 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
4		nbgrcl.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	eqtr4i 2795	. . . 4 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = 𝑉
6	5	eleq2i 2861	. . 3 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) ↔ 𝑋 ∈ 𝑉)
7	6	biimpi 219	. 2 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → 𝑋 ∈ 𝑉)
8	2, 7	simpl2im 512	1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  {crab 3423  Vcvv 3463  ⦋csb 3861   ∖ cdif 3910   ⊆ wss 3913  {csn 4594  {cpr 4596  ‘cfv 6537  (class class class)co 7411  Vtxcvtx 29286  Edgcedg 29337   NeighbVtx cnbgr 29622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-nbgr 29623

This theorem is referenced by:  nbgrel  29630  frgrnbnb  30584  pgnbgreunbgrlem3  48771  pgnbgreunbgrlem6  48777