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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sclnbgrelself | Structured version Visualization version GIF version | ||
| Description: A vertex 𝑁 is a member of its semiclosed neighborhood iff there is an edge joining the vertex with a vertex. (Contributed by AV, 16-May-2025.) |
| Ref | Expression |
|---|---|
| dfsclnbgr2.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| dfsclnbgr2.s | ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} |
| dfsclnbgr2.e | ⊢ 𝐸 = (Edg‘𝐺) |
| Ref | Expression |
|---|---|
| sclnbgrelself | ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfsclnbgr2.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | dfsclnbgr2.s | . . 3 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} | |
| 3 | dfsclnbgr2.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
| 4 | 1, 2, 3 | sclnbgrel 47833 | . 2 ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒)) |
| 5 | dfsn2 4639 | . . . . . . 7 ⊢ {𝑁} = {𝑁, 𝑁} | |
| 6 | 5 | eqcomi 2746 | . . . . . 6 ⊢ {𝑁, 𝑁} = {𝑁} |
| 7 | 6 | sseq1i 4012 | . . . . 5 ⊢ ({𝑁, 𝑁} ⊆ 𝑒 ↔ {𝑁} ⊆ 𝑒) |
| 8 | snssg 4783 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑒 ↔ {𝑁} ⊆ 𝑒)) | |
| 9 | 7, 8 | bitr4id 290 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ({𝑁, 𝑁} ⊆ 𝑒 ↔ 𝑁 ∈ 𝑒)) |
| 10 | 9 | rexbidv 3179 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒)) |
| 11 | 10 | pm5.32i 574 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒) ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒)) |
| 12 | 4, 11 | bitri 275 | 1 ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ⊆ wss 3951 {csn 4626 {cpr 4628 ‘cfv 6561 Vtxcvtx 29013 Edgcedg 29064 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: (None) |
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