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Mirrors > Home > MPE Home > Th. List > nbgrssovtx | Structured version Visualization version GIF version |
Description: The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself. Stronger version of nbgrssvtx 27402. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.) |
Ref | Expression |
---|---|
nbgrssovtx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbgrssovtx | ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrssovtx.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | nbgrisvtx 27401 | . . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ 𝑉) |
3 | nbgrnself2 27420 | . . . . 5 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋) | |
4 | df-nel 3040 | . . . . . 6 ⊢ (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋)) | |
5 | neleq1 3044 | . . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋))) | |
6 | 4, 5 | bitr3id 288 | . . . . 5 ⊢ (𝑣 = 𝑋 → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋))) |
7 | 3, 6 | mpbiri 261 | . . . 4 ⊢ (𝑣 = 𝑋 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋)) |
8 | 7 | necon2ai 2964 | . . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ≠ 𝑋) |
9 | eldifsn 4690 | . . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑋)) | |
10 | 2, 8, 9 | sylanbrc 586 | . 2 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ (𝑉 ∖ {𝑋})) |
11 | 10 | ssriv 3895 | 1 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∉ wnel 3039 ∖ cdif 3854 ⊆ wss 3857 {csn 4531 ‘cfv 6369 (class class class)co 7202 Vtxcvtx 27059 NeighbVtx cnbgr 27392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-nbgr 27393 |
This theorem is referenced by: nbgrssvwo2 27422 nbfusgrlevtxm1 27437 uvtxnbgr 27460 nbusgrvtxm1uvtx 27465 nbupgruvtxres 27467 |
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