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Mirrors > Home > MPE Home > Th. List > uvtxnbgrvtx | Structured version Visualization version GIF version |
Description: A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
Ref | Expression |
---|---|
uvtxel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
uvtxnbgrvtx | ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvtxel.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | vtxnbuvtx 27172 | . . 3 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) |
3 | eleq1w 2895 | . . . . . . 7 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))) | |
4 | 3 | rspcva 3620 | . . . . . 6 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) |
5 | nbgrsym 27144 | . . . . . . 7 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∈ (𝐺 NeighbVtx 𝑣)) | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∈ (𝐺 NeighbVtx 𝑣))) |
7 | 4, 6 | syl5ibcom 247 | . . . . 5 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) → (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣))) |
8 | 7 | expcom 416 | . . . 4 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))) |
9 | 8 | com23 86 | . . 3 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁) → (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))) |
10 | 2, 9 | mpcom 38 | . 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣))) |
11 | 10 | ralrimiv 3181 | 1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∖ cdif 3932 {csn 4566 ‘cfv 6354 (class class class)co 7155 Vtxcvtx 26780 NeighbVtx cnbgr 27113 UnivVtxcuvtx 27166 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-nbgr 27114 df-uvtx 27167 |
This theorem is referenced by: (None) |
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