Theorem nbgrnself 27629

Description: A vertex in a graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 21-Mar-2021.)

Hypothesis

Ref	Expression
nbgrnself.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbgrnself	⊢ ∀𝑣 ∈ 𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)

Distinct variable group:   𝑣,𝑉

Allowed substitution hint:   𝐺(𝑣)

Proof of Theorem nbgrnself

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

Step	Hyp	Ref	Expression
1		neldifsnd 4723	. . . . 5 ⊢ (𝑣 ∈ 𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
2	1	intnanrd 489	. . . 4 ⊢ (𝑣 ∈ 𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3		df-nel 3049	. . . . 5 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4		preq2 4667	. . . . . . . 8 ⊢ (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
5	4	sseq1d 3948	. . . . . . 7 ⊢ (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
6	5	rexbidv 3225	. . . . . 6 ⊢ (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
7	6	elrab 3617	. . . . 5 ⊢ (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
8	3, 7	xchbinx 333	. . . 4 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
9	2, 8	sylibr 233	. . 3 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
10		eqidd 2739	. . . 4 ⊢ (𝑣 ∈ 𝑉 → 𝑣 = 𝑣)
11		nbgrnself.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
12		eqid 2738	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13	11, 12	nbgrval 27606	. . . 4 ⊢ (𝑣 ∈ 𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
14	10, 13	neleq12d 3052	. . 3 ⊢ (𝑣 ∈ 𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
15	9, 14	mpbird 256	. 2 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∉ (𝐺 NeighbVtx 𝑣))
16	15	rgen 3073	1 ⊢ ∀𝑣 ∈ 𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∉ wnel 3048  ∀wral 3063  ∃wrex 3064  {crab 3067   ∖ 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

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-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  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-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-nbgr 27603

This theorem is referenced by:  nbgrnself2  27630