Theorem nbgrcl 29262

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 29260	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
2	1	mpoxeldm 8190	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ V ∧ 𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
3		csbfv 6908	. . . . 5 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
4		nbgrcl.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	eqtr4i 2755	. . . 4 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = 𝑉
6	5	eleq2i 2820	. . 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 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3405  Vcvv 3447  ⦋csb 3862   ∖ cdif 3911   ⊆ wss 3914  {csn 4589  {cpr 4591  ‘cfv 6511  (class class class)co 7387  Vtxcvtx 28923  Edgcedg 28974   NeighbVtx cnbgr 29259

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-nbgr 29260

This theorem is referenced by:  nbgrel  29267  frgrnbnb  30222  pgnbgreunbgrlem3  48108  pgnbgreunbgrlem6  48114