Theorem nbgrnself 29446

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 4726	. . . . 5 ⊢ (𝑣 ∈ 𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
2	1	intnanrd 490	. . . 4 ⊢ (𝑣 ∈ 𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3		df-nel 3039	. . . . 5 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4		preq2 4666	. . . . . . . 8 ⊢ (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
5	4	sseq1d 3946	. . . . . . 7 ⊢ (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
6	5	rexbidv 3163	. . . . . 6 ⊢ (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
7	6	elrab 3629	. . . . 5 ⊢ (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
8	3, 7	xchbinx 335	. . . 4 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
9	2, 8	sylibr 235	. . 3 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
10		eqidd 2740	. . . 4 ⊢ (𝑣 ∈ 𝑉 → 𝑣 = 𝑣)
11		nbgrnself.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
12		eqid 2739	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13	11, 12	nbgrval 29423	. . . 4 ⊢ (𝑣 ∈ 𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
14	10, 13	neleq12d 3043	. . 3 ⊢ (𝑣 ∈ 𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
15	9, 14	mpbird 258	. 2 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∉ (𝐺 NeighbVtx 𝑣))
16	15	rgen 3055	1 ⊢ ∀𝑣 ∈ 𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∉ wnel 3038  ∀wral 3053  ∃wrex 3063  {crab 3391   ∖ cdif 3880   ⊆ wss 3883  {csn 4555  {cpr 4557  ‘cfv 6485  (class class class)co 7356  Vtxcvtx 29083  Edgcedg 29134   NeighbVtx cnbgr 29419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-nbgr 29420

This theorem is referenced by:  nbgrnself2  29447