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Mathbox for Alexander van der Vekens |
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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 2735 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | dfclnbgr4 47749 | . . 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 29389 | . . . 4 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅) | |
5 | 4 | adantr 480 | . . 3 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝐾) = ∅) |
6 | 5 | uneq2d 4178 | . 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)) = ({𝐾} ∪ ∅)) |
7 | un0 4400 | . . 3 ⊢ ({𝐾} ∪ ∅) = {𝐾} | |
8 | 7 | a1i 11 | . 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ ∅) = {𝐾}) |
9 | 3, 6, 8 | 3eqtrd 2779 | 1 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ∅c0 4339 {csn 4631 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 Edgcedg 29079 NeighbVtx cnbgr 29364 ClNeighbVtx cclnbgr 47743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-nbgr 29365 df-clnbgr 47744 |
This theorem is referenced by: (None) |
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