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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 26281. (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 26280 | . . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ 𝑉) |
3 | nbgrnself2 26301 | . . . . 5 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋) | |
4 | df-nel 2927 | . . . . . 6 ⊢ (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋)) | |
5 | neleq1 2931 | . . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋))) | |
6 | 4, 5 | syl5bbr 274 | . . . . 5 ⊢ (𝑣 = 𝑋 → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋))) |
7 | 3, 6 | mpbiri 248 | . . . 4 ⊢ (𝑣 = 𝑋 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋)) |
8 | 7 | necon2ai 2852 | . . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ≠ 𝑋) |
9 | eldifsn 4350 | . . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑋)) | |
10 | 2, 8, 9 | sylanbrc 699 | . 2 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ (𝑉 ∖ {𝑋})) |
11 | 10 | ssriv 3640 | 1 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∉ wnel 2926 ∖ cdif 3604 ⊆ wss 3607 {csn 4210 ‘cfv 5926 (class class class)co 6690 Vtxcvtx 25919 NeighbVtx cnbgr 26269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-nbgr 26270 |
This theorem is referenced by: nbgrssvwo2 26303 nbfusgrlevtxm1 26323 uvtxnbgr 26351 nbusgrvtxm1uvtx 26356 nbupgruvtxres 26358 |
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