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| Mirrors > Home > MPE Home > Th. List > Mathboxes > clnbgrssedg | Structured version Visualization version GIF version | ||
| Description: The vertices connected by an edge are a subset of the neighborhood of each of these vertices. (Contributed by AV, 26-May-2025.) (Proof shortened by AV, 24-Aug-2025.) |
| Ref | Expression |
|---|---|
| clnbgrssedg.e | ⊢ 𝐸 = (Edg‘𝐺) |
| clnbgrssedg.n | ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋) |
| Ref | Expression |
|---|---|
| clnbgrssedg | ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clnbgrssedg.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
| 2 | clnbgrssedg.n | . . . . 5 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋) | |
| 3 | 1, 2 | clnbgredg 47799 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾)) → 𝑣 ∈ 𝑁) |
| 4 | 3 | 3exp2 1355 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ 𝐸 → (𝑋 ∈ 𝐾 → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁)))) |
| 5 | 4 | 3imp 1111 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁)) |
| 6 | 5 | ssrdv 3988 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3950 ‘cfv 6559 (class class class)co 7429 Edgcedg 29054 UHGraphcuhgr 29063 ClNeighbVtx cclnbgr 47778 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-1st 8010 df-2nd 8011 df-edg 29055 df-uhgr 29065 df-clnbgr 47779 |
| This theorem is referenced by: grlimgrtrilem1 47934 |
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