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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sclnbgrisvtx | Structured version Visualization version GIF version | ||
| Description: Every member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 is 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 | 
|---|---|
| sclnbgrisvtx | ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝑉) | 
| 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 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒)) | 
| 5 | 4 | simplbi 497 | 1 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝑉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ⊆ wss 3951 {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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