Theorem nbgrnself 27143

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 4728	. . . . 5 ⊢ (𝑣 ∈ 𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
2	1	intnanrd 492	. . . 4 ⊢ (𝑣 ∈ 𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3		df-nel 3126	. . . . 5 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4		preq2 4672	. . . . . . . 8 ⊢ (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
5	4	sseq1d 4000	. . . . . . 7 ⊢ (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
6	5	rexbidv 3299	. . . . . 6 ⊢ (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
7	6	elrab 3682	. . . . 5 ⊢ (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
8	3, 7	xchbinx 336	. . . 4 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
9	2, 8	sylibr 236	. . 3 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
10		eqidd 2824	. . . 4 ⊢ (𝑣 ∈ 𝑉 → 𝑣 = 𝑣)
11		nbgrnself.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
12		eqid 2823	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13	11, 12	nbgrval 27120	. . . 4 ⊢ (𝑣 ∈ 𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
14	10, 13	neleq12d 3129	. . 3 ⊢ (𝑣 ∈ 𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
15	9, 14	mpbird 259	. 2 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∉ (𝐺 NeighbVtx 𝑣))
16	15	rgen 3150	1 ⊢ ∀𝑣 ∈ 𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∉ wnel 3125  ∀wral 3140  ∃wrex 3141  {crab 3144   ∖ cdif 3935   ⊆ wss 3938  {csn 4569  {cpr 4571  ‘cfv 6357  (class class class)co 7158  Vtxcvtx 26783  Edgcedg 26834   NeighbVtx cnbgr 27116

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-nbgr 27117

This theorem is referenced by:  nbgrnself2  27144