Theorem nbgrnself 26257

Description: A vertex in a graph is not a neighbor of itself. (Contributed by by AV, 3-Nov-2020.) (Revised by AV, 21-Mar-2021.)

Hypothesis

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

Assertion

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

Distinct variable group:   𝑣,𝐺

Allowed substitution hint:   𝑉(𝑣)

Proof of Theorem nbgrnself

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

Step	Hyp	Ref	Expression
1		neldifsnd 4322	. . . . 5 ⊢ (𝑣 ∈ 𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
2	1	intnanrd 963	. . . 4 ⊢ (𝑣 ∈ 𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3		df-nel 2898	. . . . 5 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4		preq2 4269	. . . . . . . 8 ⊢ (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
5	4	sseq1d 3632	. . . . . . 7 ⊢ (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
6	5	rexbidv 3052	. . . . . 6 ⊢ (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
7	6	elrab 3363	. . . . 5 ⊢ (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
8	3, 7	xchbinx 324	. . . 4 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
9	2, 8	sylibr 224	. . 3 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
10		eqidd 2623	. . . 4 ⊢ (𝑣 ∈ 𝑉 → 𝑣 = 𝑣)
11		nbgrisvtx.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
12		eqid 2622	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13	11, 12	nbgrval 26234	. . . 4 ⊢ (𝑣 ∈ 𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
14	10, 13	neleq12d 2901	. . 3 ⊢ (𝑣 ∈ 𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
15	9, 14	mpbird 247	. 2 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∉ (𝐺 NeighbVtx 𝑣))
16	15	rgen 2922	1 ⊢ ∀𝑣 ∈ 𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  ∀wral 2912  ∃wrex 2913  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  {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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-nbgr 26228

This theorem is referenced by:  usgrnbnself  26258  nbgrnself2  26259