Theorem nbgrnvtx0 26237

Description: There are no neighbors of a class which is not a vertex. (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	⊢ (𝑁 ∉ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)

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 6233	. . . . . 6 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
3	1, 2	eqtr4i 2647	. . . . 5 ⊢ 𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)
4		neleq2 2903	. . . . 5 ⊢ (𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → (𝑁 ∉ 𝑉 ↔ 𝑁 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝑁 ∉ 𝑉 ↔ 𝑁 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))
6	5	biimpi 206	. . 3 ⊢ (𝑁 ∉ 𝑉 → 𝑁 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))
7	6	olcd 408	. 2 ⊢ (𝑁 ∉ 𝑉 → (𝐺 ∉ V ∨ 𝑁 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
8		df-nbgr 26228	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
9	8	mpt2xneldm 7338	. 2 ⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) → (𝐺 NeighbVtx 𝑁) = ∅)
10	7, 9	syl 17	1 ⊢ (𝑁 ∉ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   = wceq 1483   ∉ wnel 2897  ∃wrex 2913  {crab 2916  Vcvv 3200  ⦋csb 3533   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ‘cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   NeighbVtx cnbgr 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-nbgr 26228

This theorem is referenced by:  nbuhgr  26239  nbumgr  26243  nbgr0vtxlem  26251  nbgr1vtx  26254