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Mathbox for Alexander van der Vekens |
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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 47771 | . 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 1537 ∈ wcel 2106 ∃wrex 3068 {crab 3433 ⊆ wss 3963 {cpr 4633 ‘cfv 6563 Vtxcvtx 29028 Edgcedg 29079 |
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-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 df-rab 3434 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 |
This theorem is referenced by: (None) |
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