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| Mirrors > Home > MPE Home > Th. List > nbgr0vtx | Structured version Visualization version GIF version | ||
| Description: In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) (Proof shortened by AV, 10-May-2025.) | 
| Ref | Expression | 
|---|---|
| nbgr0vtx | ⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nel02 4339 | . . 3 ⊢ ((Vtx‘𝐺) = ∅ → ¬ 𝐾 ∈ (Vtx‘𝐺)) | |
| 2 | df-nel 3047 | . . 3 ⊢ (𝐾 ∉ (Vtx‘𝐺) ↔ ¬ 𝐾 ∈ (Vtx‘𝐺)) | |
| 3 | 1, 2 | sylibr 234 | . 2 ⊢ ((Vtx‘𝐺) = ∅ → 𝐾 ∉ (Vtx‘𝐺)) | 
| 4 | eqid 2737 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 5 | 4 | nbgrnvtx0 29356 | . 2 ⊢ (𝐾 ∉ (Vtx‘𝐺) → (𝐺 NeighbVtx 𝐾) = ∅) | 
| 6 | 3, 5 | syl 17 | 1 ⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Vtxcvtx 29013 NeighbVtx cnbgr 29349 | 
| 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-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-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 | 
| This theorem is referenced by: (None) | 
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