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| Mirrors > Home > MPE Home > Th. List > Mathboxes > clnbgr0edg | Structured version Visualization version GIF version | ||
| Description: In an empty graph (with no edges), all closed neighborhoods consists of a single vertex. (Contributed by AV, 10-May-2025.) | 
| Ref | Expression | 
|---|---|
| clnbgr0edg | ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 2 | 1 | dfclnbgr4 47811 | . . 3 ⊢ (𝐾 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝐾) = ({𝐾} ∪ (𝐺 NeighbVtx 𝐾))) | 
| 3 | 2 | adantl 481 | . 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = ({𝐾} ∪ (𝐺 NeighbVtx 𝐾))) | 
| 4 | nbgr0edg 29374 | . . . 4 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅) | |
| 5 | 4 | adantr 480 | . . 3 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝐾) = ∅) | 
| 6 | 5 | uneq2d 4168 | . 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)) = ({𝐾} ∪ ∅)) | 
| 7 | un0 4394 | . . 3 ⊢ ({𝐾} ∪ ∅) = {𝐾} | |
| 8 | 7 | a1i 11 | . 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ ∅) = {𝐾}) | 
| 9 | 3, 6, 8 | 3eqtrd 2781 | 1 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∅c0 4333 {csn 4626 ‘cfv 6561 (class class class)co 7431 Vtxcvtx 29013 Edgcedg 29064 NeighbVtx cnbgr 29349 ClNeighbVtx cclnbgr 47805 | 
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-nbgr 29350 df-clnbgr 47806 | 
| This theorem is referenced by: (None) | 
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