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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfnbgrss2 | Structured version Visualization version GIF version | ||
| Description: Subset chain for different kinds of neighborhoods of a vertex. (Contributed by AV, 16-May-2025.) | 
| Ref | Expression | 
|---|---|
| dfvopnbgr2.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| dfvopnbgr2.e | ⊢ 𝐸 = (Edg‘𝐺) | 
| dfvopnbgr2.u | ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))} | 
| dfsclnbgr6.s | ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} | 
| Ref | Expression | 
|---|---|
| dfnbgrss2 | ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfvopnbgr2.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | dfvopnbgr2.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
| 3 | dfvopnbgr2.u | . . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))} | |
| 4 | 1, 2, 3 | dfnbgr6 47843 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁})) | 
| 5 | difss 4136 | . . 3 ⊢ (𝑈 ∖ {𝑁}) ⊆ 𝑈 | |
| 6 | 4, 5 | eqsstrdi 4028 | . 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑈) | 
| 7 | ssun1 4178 | . . 3 ⊢ 𝑈 ⊆ (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}) | |
| 8 | dfsclnbgr6.s | . . . 4 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} | |
| 9 | 1, 2, 3, 8 | dfsclnbgr6 47844 | . . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒})) | 
| 10 | 7, 9 | sseqtrrid 4027 | . 2 ⊢ (𝑁 ∈ 𝑉 → 𝑈 ⊆ 𝑆) | 
| 11 | 1, 8, 2 | dfnbgrss 47838 | . . 3 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁))) | 
| 12 | 11 | simprd 495 | . 2 ⊢ (𝑁 ∈ 𝑉 → 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)) | 
| 13 | 6, 10, 12 | 3jca 1129 | 1 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 ‘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-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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