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Mirrors > Home > MPE Home > Th. List > nbgrnself | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nbgrnself.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbgrnself | ⊢ ∀𝑣 ∈ 𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neldifsnd 4795 | . . . . 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 3047 | . . . . 5 ⊢ (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}) | |
4 | preq2 4737 | . . . . . . . 8 ⊢ (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣}) | |
5 | 4 | sseq1d 4012 | . . . . . . 7 ⊢ (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒)) |
6 | 5 | rexbidv 3178 | . . . . . 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 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 2733 | . . . 4 ⊢ (𝑣 ∈ 𝑉 → 𝑣 = 𝑣) | |
11 | nbgrnself.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | eqid 2732 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
13 | 11, 12 | nbgrval 28582 | . . . 4 ⊢ (𝑣 ∈ 𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}) |
14 | 10, 13 | neleq12d 3051 | . . 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 3063 | 1 ⊢ ∀𝑣 ∈ 𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∉ wnel 3046 ∀wral 3061 ∃wrex 3070 {crab 3432 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 {cpr 4629 ‘cfv 6540 (class class class)co 7405 Vtxcvtx 28245 Edgcedg 28296 NeighbVtx cnbgr 28578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-nbgr 28579 |
This theorem is referenced by: nbgrnself2 28606 |
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