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Mirrors > Home > MPE Home > Th. List > nbgrisvtx | Structured version Visualization version GIF version |
Description: Every neighbor 𝑁 of a vertex 𝐾 is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.) |
Ref | Expression |
---|---|
nbgrisvtx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbgrisvtx | ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrisvtx.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2736 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | nbgrel 28174 | . 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)) |
4 | simp1l 1197 | . 2 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁 ∈ 𝑉) | |
5 | 3, 4 | sylbi 216 | 1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∃wrex 3071 ⊆ wss 3908 {cpr 4586 ‘cfv 6493 (class class class)co 7353 Vtxcvtx 27833 Edgcedg 27884 NeighbVtx cnbgr 28166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-nbgr 28167 |
This theorem is referenced by: nbgrssvtx 28176 nbgrnself2 28194 nbgrssovtx 28195 frgrnbnb 29123 frgrncvvdeqlem2 29130 frgrncvvdeqlem3 29131 frgrncvvdeqlem9 29137 numclwwlk1lem2foa 29184 numclwwlk1lem2fo 29188 |
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