Theorem nbgrnvtx0 28864

Description: If a class 𝑋 is not a vertex of a graph 𝐺, then it has no neighbors in 𝐺. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)

Hypothesis

Ref	Expression
nbgrel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbgrnvtx0	⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅)

Proof of Theorem nbgrnvtx0

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

Step	Hyp	Ref	Expression
1		nbgrel.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2		csbfv 6941	. . . . . 6 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
3	1, 2	eqtr4i 2762	. . . . 5 ⊢ 𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)
4		neleq2 3052	. . . . 5 ⊢ (𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → (𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))
6	5	biimpi 215	. . 3 ⊢ (𝑋 ∉ 𝑉 → 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))
7	6	olcd 871	. 2 ⊢ (𝑋 ∉ 𝑉 → (𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
8		df-nbgr 28858	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
9	8	mpoxneldm 8200	. 2 ⊢ ((𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) → (𝐺 NeighbVtx 𝑋) = ∅)
10	7, 9	syl 17	1 ⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅)

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 844   = wceq 1540   ∉ wnel 3045  ∃wrex 3069  {crab 3431  Vcvv 3473  ⦋csb 3893   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  {csn 4628  {cpr 4630  ‘cfv 6543  (class class class)co 7412  Vtxcvtx 28524  Edgcedg 28575   NeighbVtx cnbgr 28857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-nbgr 28858

This theorem is referenced by:  nbuhgr  28868  nbumgr  28872  nbgr0vtxlem  28880  nbgr1vtx  28883